Start the analysis of a budget error

index.org

@@ -1857,6 +1857,7 @@ The file =mat/plant.mat= is accessible [[./mat/plant.mat][here]].
 #+end_src
 
 * Huddle Test
+** Introduction                                                     :ignore:
 ** Matlab Init                                              :noexport:ignore:
 #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
   <<matlab-dir>>
@@ -1960,16 +1961,274 @@ We compute the Power Spectral Density of the voltage across the inductance used
   xlim([1, 1000]);
 #+end_src
 
+* Budget Error
+** Introduction                                                     :ignore:
+Goals:
+- List all sources of error and their effects on the Attocube measurement
+- Think about how to determine the value of the individual sources of error
+
+Sources of error for the Attocube measurement:
+- Cercalo rotation error
+- Cercalo unwanted translation perpendicular to its surface
+- Newport rotation error
+- Newport unwanted translation perpendicular to its surface
+- Temperature variations of the metrology frame
+- Temperature / Pressure / Humidity variations of the air in the beam path
+
+** Matlab Init                                              :noexport:ignore:
+#+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
+
+** Effect of the Cercalo angle error on the measured distance by the Attocube
+To simplify, we suppose that the Newport mirror is a flat mirror.
+The geometry of the setup is shown in Fig. [[fig:angle_error_schematic_cercalo]].
+
+We define the angle error range where we want to evaluate the distance error measured by the Attocube.
+#+begin_src matlab
+  thetas_c = logspace(-7, -4, 100); % [rad]
+#+end_src
+
+The geometrical parameters of the setup are defined below.
+#+begin_src matlab
+  H = 0.05; % [m]
+  L = 0.05; % [m]
+#+end_src
+
+#+begin_src latex :file angle_error_schematic_cercalo.pdf :exports results
+  \begin{tikzpicture}
+    \draw[->] (0, 0)node[branch](O){}node[below]{$O (0,0)$} -- ++(1, 0) node[above left]{$x$};
+    \draw[->] (0, 0) -- ++(0, 1) node[below right]{$y$};
+    \draw[] (4, 0)node[branch](S){}node[below right]{$S (L,0)$} -- ++(45:1);
+    \draw[] (4, 0) -- ++(225:1);
+    \draw[] (3, 2) --node[midway, branch](X){}node[above]{$X (0,H)$} (5, 2);
+    \draw[<->] ([shift=(30:1.2)]S.center) arc (30:60:1.2) node[midway, above right]{$\theta_c$};
+
+    \draw[red, ->-=.7, -<-=0.3] (O.center) -- (S.center);
+    \draw[red, ->-=.7, -<-=0.3] (S.center) -- (X.center);
+  \end{tikzpicture}
+#+end_src
+
+#+NAME: fig:angle_error_schematic_cercalo
+#+CAPTION: Caption
+#+RESULTS:
+[[file:figs/angle_error_schematic_cercalo.png]]
+
+The nominal points $O$, $S$ and $X$ are defined.
+#+begin_src matlab
+  O = [-L, 0];
+  S = [0, 0];
+  X = [0, H];
+#+end_src
+
+Thus, the initial path length is:
+#+begin_src matlab
+  path_nominal = norm(S-O) + norm(X-S) + norm(S-X) + norm(O-S);
+#+end_src
+
+We now compute the new path length when there is an error angle $\delta \theta_c$ of the Cercalo.
+#+begin_src matlab
+  path_length = zeros(size(thetas_c));
+
+  for i = 1:length(thetas_c)
+      theta_c = thetas_c(i);
+      Y = [H*tan(2*theta_c), H];
+      M = 2*H/(tan(pi/4-theta_c)+1/tan(2*theta_c))*[1, tan(pi/4-theta_c)];
+      T = [-L, M(2)+(L+M(1))*tan(4*theta_c)];
+
+      path_length(i) = norm(S-O) + norm(Y-S) + norm(M-Y) + norm(T-M);
+  end
+#+end_src
+
+We then compute the distance error and we plot it as a function of the Cercalo angle error (Fig. [[]]).
+#+begin_src matlab
+  path_error = path_length - path_nominal;
+#+end_src
+
+#+begin_src matlab :exports none
+  figure;
+  plot(thetas_c, path_error)
+  set(gca,'xscale','log');
+  set(gca,'yscale','log');
+  xlabel('Cercalo angle error [rad]');
+  ylabel('Attocube measurement error [m]');
+#+end_src
+
+#+HEADER: :tangle no :exports results :results none :noweb yes
+#+begin_src matlab :var filepath="figs/effect_cercalo_angle_distance_meas.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
+  <<plt-matlab>>
+#+end_src
+
+#+NAME: fig:effect_cercalo_angle_distance_meas
+#+CAPTION: Effect of an angle error of the Cercalo on the distance error measured by the Attocube ([[./figs/effect_cercalo_angle_distance_meas.png][png]], [[./figs/effect_cercalo_angle_distance_meas.pdf][pdf]])
+[[file:figs/effect_cercalo_angle_distance_meas.png]]
+
+And we plot the beam path using Matlab for an high angle to verify that the code is working (Fig. [[]]).
+#+begin_src matlab
+  theta = 2*2*pi/360; % [rad]
+  H = 0.05; % [m]
+  L = 0.05; % [m]
+
+  O = [-L, 0];
+  S = [0, 0];
+  X = [0, H];
+  Y = [H*tan(2*theta), H];
+  M = 2*H/(tan(pi/4-theta)+1/tan(2*theta))*[1, tan(pi/4-theta)];
+  T = [-L, M(2)+(L+M(1))*tan(4*theta)];
+#+end_src
+
+#+begin_src matlab :exports none
+  figure;
+  hold on;
+  plot([-L, -L], [0, H], 'k-'); % Interferometer
+  plot([-L, 0.1*L], [H, H], 'k-'); % Reflector
+  plot(0.5*min(L, H)*[-cos(pi/4-theta), cos(pi/4-theta)], 0.5*min(L, H)*[-sin(pi/4-theta), sin(pi/4-theta)], 'k-'); % Tilt-Mirror
+  plot(0.5*min(L, H)*[-cos(pi/4), cos(pi/4)], 0.5*min(L, H)*[-sin(pi/4), sin(pi/4)], 'k--'); % Initial position of tilt mirror
+  plot([O(1), S(1), Y(1), M(1), T(1)], [O(2), S(2), Y(2), M(2), T(2)], 'r-');
+  plot([O(1), S(1), X(1), S(1), O(1)], [O(2), S(2), X(2), S(2), O(2)], 'b--');
+  hold off;
+  xlabel('X [m]'); ylabel('Y [m]');
+  axis equal
+#+end_src
+
+#+HEADER: :tangle no :exports results :results none :noweb yes
+#+begin_src matlab :var filepath="figs/simulation_beam_path_high_angle.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
+  <<plt-matlab>>
+#+end_src
+
+#+NAME: fig:simulation_beam_path_high_angle
+#+CAPTION: Simulation of a beam path for high angle error ([[./figs/simulation_beam_path_high_angle.png][png]], [[./figs/simulation_beam_path_high_angle.pdf][pdf]])
+[[file:figs/simulation_beam_path_high_angle.png]]
+
+#+begin_important
+  An angle error $\delta\theta_c$ of the Attocube produces a distance error measured by the attocube of
+  \begin{equation}
+    \delta L = 10^{-6} \cdot \delta\theta_c
+  \end{equation}
+  Thus, $1 \mu \text{rad}$ of angle error corresponds to $1mn$ of distance error.
+#+end_important
+
+** Estimation of the Cercalo angle error due to Noise
+*** Perfect Control
+If the feedback is perfect, the Cercalo angle error will be equal to the 4 quadrant diode noise.
+Let's estimate the 4QD noise in radians.
+
+If we note $V_1$, $V_2$, $V_3$ and $V_4$ the voltage of each of the quadrant, a measurement error $\delta V_i$ of one of the quadrant will have an effect $\delta \theta$ on the measured angle:
+\[ \delta\theta = G \frac{\delta V_i}{V_1 + V_2 + V_3 + V_4} \]
+with $G$ is the gain of the 4QD in [rad].
+
+We should then have that the voltage of each quadrant is as large as possible.
+Suppose here that $V_i \approx 5V$, $\delta V_i = 1mV$ and $G = 0.03\,rad$, we obtain:
+\[ \delta \theta = 0.03 \frac{0.001}{20} = 1.5\, \mu\text{rad} \]
+This then corresponds to
+\[ \delta L = 10^{-6} \cdot \delta \theta = 1.5\,nm \]
+
+If we just consider the ADC noise:
+- the ADC range is $\pm 10V$ with $16\text{ bits}$.
+- thus, the LSB corresponds to:
+  \[ \frac{20}{2^{16}} \approx 0.000305\,V = 0.305\,mV \]
+- this corresponds to an error $\delta L \approx 0.5 nm$
+
+*** Error due to DAC noise used for the Cercalo
+#+begin_src matlab
+  load('./mat/plant.mat', 'Gi', 'Gc', 'Gd');
+#+end_src
+
+#+begin_src matlab
+  G = inv(Gd)*Gc*Gi;
+#+end_src
+
+Dynamical estimation:
+- ASD of DAC noise used for the Cercalo
+- Multiply by transfer function from Cercalo voltage to angle estimation using the 4QD
+
+#+begin_src matlab
+  freqs = logspace(1, 3, 1000);
+
+  fs = 1e4;
+
+  % ASD of the DAC voltage going to the Cercalo in [V/sqrt(Hz)]
+  asd_uc = (20/2^16)/sqrt(12*fs)*ones(length(freqs), 1);
+
+  % ASD of the measured angle by the QD in [rad/sqrt(Hz)]
+  asd_theta = asd_uc.*abs(squeeze(freqresp(G(1,1), freqs, 'Hz')));
+
+  figure;
+  loglog(freqs, asd_theta)
+#+end_src
+
+Then the corresponding ASD of the measured displacement by the interferometer is:
+#+begin_src matlab
+  asd_L = asd_theta*10^(-6); % [m/sqrt(Hz)]
+#+end_src
+
+And we integrate that to have the RMS value:
+#+begin_src matlab
+  cps_L = 1/pi*cumtrapz(2*pi*freqs, (asd_L).^2);
+#+end_src
+
+The RMS value is:
+#+begin_src matlab :results value replace
+  sqrt(cps_L(end))
+#+end_src
+
+#+RESULTS:
+: 1.647e-11
+
+#+begin_src matlab
+  figure;
+  loglog(freqs, cps_L)
+#+end_src
+
+Let's estimate the beam angle error corresponding to 1 LSB of the cercalo's DAC.
+Gain of the Cercalo is approximatively 5 degrees for 10V.
+However the beam angle deviation is 4 times the angle deviation of the cercalo mirror, thus:
+#+begin_src matlab :results value replace
+  d_alpha = 4*(20/2^16)*(5*pi/180)/10 % [rad]
+#+end_src
+
+#+RESULTS:
+: 1.0653e-05
+
+This corresponds to a measurement error of the Attocube equals to (in [m])
+#+begin_src matlab :results value replace
+  1e-6*d_alpha % [m]
+#+end_src
+
+#+RESULTS:
+: 1.0653e-11
+
+#+begin_important
+  The DAC noise use for the Cercalo does not limit the performance of the system.
+#+end_important
+
 * Plant Scaling
-|                        | Value | Unit        |   |
-|------------------------+-------+-------------+---|
-| Expected perturbations |     1 | [V]         | $U_n$ |
-| Maximum input usage    |    10 | [V]         | $U_c$ |
+|                        | Value | Unit        |          |
+|------------------------+-------+-------------+----------|
+| Expected perturbations |     1 | [V]         | $U_n$    |
+| Maximum input usage    |    10 | [V]         | $U_c$    |
 | Maximum wanted error   |    10 | [$\mu rad$] | $\theta$ |
-| Measured noise         |     5 | [$\mu rad$] |   |
+| Measured noise         |     5 | [$\mu rad$] |          |
 
 ** General Configuration
 
+* Control Objective
+The maximum expected stroke is $y_\text{max} = 3mm \approx 5e^{-2} rad$ at $1Hz$.
+The maximum wanted error is $e_\text{max} = 10 \mu rad$.
+
+Thus, we require the sensitivity function at $\omega_0 = 1\text{ Hz}$:
+\begin{align*}
+  |S(j\omega_0)| &< \left| \frac{e_\text{max}}{y_\text{max}} \right| \\
+                 &< 2 \cdot 10^{-4}
+\end{align*}
+
+In terms of loop gain, this is equivalent to:
+\[ |L(j\omega_0)| > 5 \cdot 10^{3} \]
+
 * Plant Analysis
 ** Matlab Init                                              :noexport:ignore:
 #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
@@ -2023,20 +2282,7 @@ We compute the Power Spectral Density of the voltage across the inductance used
   G0 = freqresp(G, 0);
 #+end_src
 
-* Control Objective
-The maximum expected stroke is $y_\text{max} = 3mm \approx 5e^{-2} rad$ at $1Hz$.
-The maximum wanted error is $e_\text{max} = 10 \mu rad$.
-
-Thus, we require the sensitivity function at $\omega_0 = 1\text{ Hz}$:
-\begin{align*}
-  |S(j\omega_0)| &< \left| \frac{e_\text{max}}{y_\text{max}} \right| \\
-                 &< 2 \cdot 10^{-4}
-\end{align*}
-
-In terms of loop gain, this is equivalent to:
-\[ |L(j\omega_0)| > 5 \cdot 10^{3} \]
-
-* Decentralized Control
+* Decentralized Control of the Cercalo
 :PROPERTIES:
 :header-args:matlab+: :tangle matlab/decentralized_control.m
 :header-args:matlab+: :comments org :mkdirp yes
@@ -2249,11 +2495,9 @@ The controllers can be downloaded [[./mat/K_newport.mat][here]].
   save('mat/K_newport.mat', 'Kn', 'Knd');
 #+end_src
 
-* Measuement of the non-repeatability
+* Measurement of the non-repeatability
 ** Introduction                                                     :ignore:
 - Explanation of the procedure
-- List all sources of error and their effects on the Attocube measurement
-- Think about how to determine the value of the individual sources of error
 
 ** Matlab Init                                              :noexport:ignore:
 #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)