Add analysis for voice coil actuators (uniaxial model)

This commit is contained in:
Thomas Dehaeze 2019-11-04 18:17:19 +01:00
parent ae13071c6e
commit 47034f56a7
9 changed files with 872 additions and 166 deletions

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@ -47,6 +47,8 @@ The idea is to use the same model as the full Simscape Model but to restrict the
This is done in order to more easily study the system and evaluate control techniques.
* Simscape Model
<<sec:simscape_model>>
A schematic of the uniaxial model used for simulations is represented in figure [[fig:uniaxial-model-nass-flexible]].
The perturbations $w$ are:
@ -623,6 +625,7 @@ Schematics of the active damping techniques are displayed in figure [[fig:uniaxi
[[file:figs/uniaxial-model-nass-flexible-active-damping.png]]
* Undamped System
<<sec:undamped>>
** Introduction :ignore:
Let's start by study the undamped system.
@ -2184,6 +2187,7 @@ And initialize the controllers.
Direct Velocity Feedback:
#+end_important
* With Cedrat Piezo-electric Actuators
<<sec:cedrat_actuator>>
** 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>>
@ -2749,3 +2753,400 @@ It is important to note that the effect of direct forces applied to the sample a
| Sensitivity ($F_s$) | - (at low freq) | + | + |
| Sensitivity ($F_{ty,rz}$) | + | - | + |
| Overall RMS of $D$ | = | = | = |
* Voice Coil
<<sec:voice_coil>>
** 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
#+begin_src matlab :tangle no
simulinkproject('../');
#+end_src
#+begin_src matlab
open 'simscape/sim_nano_station_uniaxial.slx'
#+end_src
** Init
We initialize all the stages with the default parameters.
The nano-hexapod is an hexapod with voice coils and the sample has a mass of 50kg.
#+begin_src matlab :exports none
initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeAxisc();
initializeMirror();
initializeNanoHexapod(struct('actuator', 'lorentz'));
initializeSample(struct('mass', 50));
#+end_src
All the controllers are set to 0 (Open Loop).
#+begin_src matlab :exports none
K = tf(0);
save('./mat/controllers.mat', 'K', '-append');
K_iff = tf(0);
save('./mat/controllers.mat', 'K_iff', '-append');
K_rmc = tf(0);
save('./mat/controllers.mat', 'K_rmc', '-append');
K_dvf = tf(0);
save('./mat/controllers.mat', 'K_dvf', '-append');
#+end_src
** Identification
We identify the dynamics of the system.
#+begin_src matlab
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;
%% Name of the Simulink File
mdl = 'sim_nano_station_uniaxial';
#+end_src
The inputs and outputs are defined below and corresponds to the name of simulink blocks.
#+begin_src matlab
%% Input/Output definition
io(1) = linio([mdl, '/Dw'], 1, 'input'); % Ground Motion
io(2) = linio([mdl, '/Fs'], 1, 'input'); % Force applied on the sample
io(3) = linio([mdl, '/Fnl'], 1, 'input'); % Force applied by the NASS
io(4) = linio([mdl, '/Fdty'], 1, 'input'); % Parasitic force Ty
io(5) = linio([mdl, '/Fdrz'], 1, 'input'); % Parasitic force Rz
io(6) = linio([mdl, '/Dsm'], 1, 'output'); % Displacement of the sample
io(7) = linio([mdl, '/Fnlm'], 1, 'output'); % Force sensor in NASS's legs
io(8) = linio([mdl, '/Dnlm'], 1, 'output'); % Displacement of NASS's legs
io(9) = linio([mdl, '/Dgm'], 1, 'output'); % Absolute displacement of the granite
io(10) = linio([mdl, '/Vlm'], 1, 'output'); % Measured absolute velocity of the top NASS platform
#+end_src
Finally, we use the =linearize= Matlab function to extract a state space model from the simscape model.
#+begin_src matlab
%% Run the linearization
G_vc = linearize(mdl, io, options);
G_vc.InputName = {'Dw', ... % Ground Motion [m]
'Fs', ... % Force Applied on Sample [N]
'Fn', ... % Force applied by NASS [N]
'Fty', ... % Parasitic Force Ty [N]
'Frz'}; % Parasitic Force Rz [N]
G_vc.OutputName = {'D', ... % Measured sample displacement x.r.t. granite [m]
'Fnm', ... % Force Sensor in NASS [N]
'Dnm', ... % Displacement Sensor in NASS [m]
'Dgm', ... % Asbolute displacement of Granite [m]
'Vlm'}; ... % Absolute Velocity of NASS [m/s]
#+end_src
Finally, we save the identified system dynamics for further analysis.
#+begin_src matlab
save('./uniaxial/mat/plants.mat', 'G_vc', '-append');
#+end_src
** Sensitivity to Disturbances
We load the dynamics when using a piezo-electric nano hexapod to compare the results.
#+begin_src matlab
load('./uniaxial/mat/plants.mat', 'G');
#+end_src
We show several plots representing the sensitivity to disturbances:
- in figure [[fig:uniaxial-sensitivity-vc-disturbances]] the transfer functions from ground motion $D_w$ to the sample position $D$ and the transfer function from direct force on the sample $F_s$ to the sample position $D$ are shown
- in figure [[fig:uniaxial-sensitivity-vc-force-dist]], it is the effect of parasitic forces of the positioning stages ($F_{ty}$ and $F_{rz}$) on the position $D$ of the sample that are shown
#+begin_src matlab :exports none
freqs = logspace(0, 3, 1000);
figure;
subplot(2, 1, 1);
title('$D_w$ to $D$');
hold on;
plot(freqs, abs(squeeze(freqresp(G_vc('D', 'Dw'), freqs, 'Hz'))), 'k-', 'DisplayName', 'G - VC');
plot(freqs, abs(squeeze(freqresp(G('D', 'Dw'), freqs, 'Hz'))), 'k--', 'DisplayName', 'G - PZ');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
legend('location', 'northeast');
ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]');
subplot(2, 1, 2);
title('$F_s$ to $D$');
hold on;
plot(freqs, abs(squeeze(freqresp(G_vc('D', 'Fs'), freqs, 'Hz'))), 'k-');
plot(freqs, abs(squeeze(freqresp(G('D', 'Fs'), freqs, 'Hz'))), 'k--');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
#+end_src
#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/uniaxial-sensitivity-vc-disturbances.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+NAME: fig:uniaxial-sensitivity-vc-disturbances
#+CAPTION: Sensitivity to disturbances ([[./figs/uniaxial-sensitivity-vc-disturbances.png][png]], [[./figs/uniaxial-sensitivity-vc-disturbances.pdf][pdf]])
[[file:figs/uniaxial-sensitivity-vc-disturbances.png]]
#+begin_src matlab :exports none
freqs = logspace(0, 3, 1000);
figure;
subplot(2, 1, 1);
title('$F_{ty}$ to $D$');
hold on;
plot(freqs, abs(squeeze(freqresp(G_vc('D', 'Fty'), freqs, 'Hz'))), 'k-', 'DisplayName', 'G - VC');
plot(freqs, abs(squeeze(freqresp(G('D', 'Fty'), freqs, 'Hz'))), 'k--', 'DisplayName', 'G - PZ');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
legend('location', 'northeast');
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
subplot(2, 1, 2);
title('$F_{rz}$ to $D$');
hold on;
plot(freqs, abs(squeeze(freqresp(G_vc('D', 'Frz'), freqs, 'Hz'))), 'k-');
plot(freqs, abs(squeeze(freqresp(G('D', 'Frz'), freqs, 'Hz'))), 'k--');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
#+end_src
#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/uniaxial-sensitivity-vc-force-dist.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+NAME: fig:uniaxial-sensitivity-vc-force-dist
#+CAPTION: Sensitivity to disturbances ([[./figs/uniaxial-sensitivity-vc-force-dist.png][png]], [[./figs/uniaxial-sensitivity-vc-force-dist.pdf][pdf]])
[[file:figs/uniaxial-sensitivity-vc-force-dist.png]]
** Noise Budget
We first load the measured PSD of the disturbance.
#+begin_src matlab
load('./disturbances/mat/dist_psd.mat', 'dist_f');
#+end_src
The effect of these disturbances on the distance $D$ is computed below.
#+begin_src matlab :exports none
% Power Spectral Density of the relative Displacement [m^2/Hz]
psd_vc_gm_d = dist_f.psd_gm.*abs(squeeze(freqresp(G_vc('D', 'Dw'), dist_f.f, 'Hz'))).^2;
psd_vc_ty_d = dist_f.psd_ty.*abs(squeeze(freqresp(G_vc('D', 'Fty'), dist_f.f, 'Hz'))).^2;
psd_vc_rz_d = dist_f.psd_rz.*abs(squeeze(freqresp(G_vc('D', 'Frz'), dist_f.f, 'Hz'))).^2;
#+end_src
#+begin_src matlab :exports none
% Power Spectral Density of the relative Displacement [m^2/Hz]
psd_gm_d = dist_f.psd_gm.*abs(squeeze(freqresp(G('D', 'Dw'), dist_f.f, 'Hz'))).^2;
psd_ty_d = dist_f.psd_ty.*abs(squeeze(freqresp(G('D', 'Fty'), dist_f.f, 'Hz'))).^2;
psd_rz_d = dist_f.psd_rz.*abs(squeeze(freqresp(G('D', 'Frz'), dist_f.f, 'Hz'))).^2;
#+end_src
The PSD of the obtain distance $D$ due to each of the perturbation is shown in figure [[fig:uniaxial-vc-psd-dist]] and the Cumulative Amplitude Spectrum is shown in figure [[fig:uniaxial-vc-cas-dist]].
The Root Mean Square value of the obtained displacement $D$ is computed below and can be determined from the figure [[fig:uniaxial-vc-cas-dist]].
#+begin_src matlab :results value replace :exports results
cas_tot_d = sqrt(cumtrapz(dist_f.f, psd_vc_rz_d+psd_vc_ty_d+psd_vc_gm_d)); cas_tot_d(end)
#+end_src
#+RESULTS:
: 4.8793e-06
#+begin_src matlab :exports none
freqs = logspace(0, 3, 1000);
figure;
hold on;
plot(dist_f.f, psd_vc_gm_d, 'DisplayName', '$D_w$');
plot(dist_f.f, psd_vc_ty_d, 'DisplayName', '$T_y$');
plot(dist_f.f, psd_vc_rz_d, 'DisplayName', '$R_z$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('CAS of the effect of disturbances on $D$ $\left[\frac{m^2}{Hz}\right]$'); xlabel('Frequency [Hz]');
legend('location', 'northeast')
xlim([0.5, 500]);
#+end_src
#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/uniaxial-vc-psd-dist.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+NAME: fig:uniaxial-vc-psd-dist
#+CAPTION: PSD of the displacement $D$ due to disturbances ([[./figs/uniaxial-vc-psd-dist.png][png]], [[./figs/uniaxial-vc-psd-dist.pdf][pdf]])
[[file:figs/uniaxial-vc-psd-dist.png]]
#+begin_src matlab :exports none
freqs = logspace(0, 3, 1000);
figure;
hold on;
plot(dist_f.f, flip(sqrt(-cumtrapz(flip(dist_f.f), flip(psd_vc_gm_d)))), 'DisplayName', '$D_w$ - VC');
plot(dist_f.f, flip(sqrt(-cumtrapz(flip(dist_f.f), flip(psd_vc_ty_d)))), 'DisplayName', '$T_y$ - VC');
plot(dist_f.f, flip(sqrt(-cumtrapz(flip(dist_f.f), flip(psd_vc_rz_d)))), 'DisplayName', '$R_z$ - VC');
plot(dist_f.f, flip(sqrt(-cumtrapz(flip(dist_f.f), flip(psd_vc_gm_d+psd_vc_ty_d+psd_vc_rz_d)))), 'k-', 'DisplayName', 'tot - VC');
plot(dist_f.f, flip(sqrt(-cumtrapz(flip(dist_f.f), flip(psd_gm_d+psd_ty_d+psd_rz_d)))), 'k--', 'DisplayName', 'tot - PZ');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('CAS of the effect of disturbances on $D$ [m]'); xlabel('Frequency [Hz]');
legend('location', 'northeast')
xlim([0.5, 500]); ylim([1e-12, 5e-6]);
#+end_src
#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/uniaxial-vc-cas-dist.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+NAME: fig:uniaxial-vc-cas-dist
#+CAPTION: CAS of the displacement $D$ due the disturbances ([[./figs/uniaxial-vc-cas-dist.png][png]], [[./figs/uniaxial-vc-cas-dist.pdf][pdf]])
[[file:figs/uniaxial-vc-cas-dist.png]]
#+begin_important
Even though the RMS value of the displacement $D$ is lower when using a piezo-electric actuator, the motion is mainly due to high frequency disturbances which are more difficult to control (an higher control bandwidth is required).
Thus, it may be desirable to use voice coil actuators.
#+end_important
** Integral Force Feedback
#+begin_src matlab
K_iff = -20/s;
#+end_src
#+begin_src matlab :exports none
freqs = logspace(-1, 2, 1000);
figure;
ax1 = subplot(2, 1, 1);
plot(freqs, abs(squeeze(freqresp(K_iff*G_vc('Fnm', 'Fn'), freqs, 'Hz'))), 'k-');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
ax2 = subplot(2, 1, 2);
plot(freqs, 180/pi*angle(squeeze(freqresp(K_iff*G_vc('Fnm', 'Fn'), freqs, 'Hz'))), 'k-');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);
linkaxes([ax1,ax2],'x');
#+end_src
#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/uniaxial_iff_vc_open_loop.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+NAME: fig:uniaxial_iff_vc_open_loop
#+CAPTION: Open Loop Transfer Function for IFF control when using a voice coil actuator ([[./figs/uniaxial_iff_vc_open_loop.png][png]], [[./figs/uniaxial_iff_vc_open_loop.pdf][pdf]])
[[file:figs/uniaxial_iff_vc_open_loop.png]]
** Identification of the Damped Plant
Let's initialize the system prior to identification.
#+begin_src matlab
initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeAxisc();
initializeMirror();
initializeNanoHexapod(struct('actuator', 'lorentz'));
initializeSample(struct('mass', 50));
#+end_src
All the controllers are set to 0.
#+begin_src matlab
K = tf(0);
save('./mat/controllers.mat', 'K', '-append');
K_iff = -K_iff;
save('./mat/controllers.mat', 'K_iff', '-append');
K_rmc = tf(0);
save('./mat/controllers.mat', 'K_rmc', '-append');
K_dvf = tf(0);
save('./mat/controllers.mat', 'K_dvf', '-append');
#+end_src
#+begin_src matlab
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;
%% Name of the Simulink File
mdl = 'sim_nano_station_uniaxial';
#+end_src
#+begin_src matlab
%% Input/Output definition
io(1) = linio([mdl, '/Dw'], 1, 'input'); % Ground Motion
io(2) = linio([mdl, '/Fs'], 1, 'input'); % Force applied on the sample
io(3) = linio([mdl, '/Fnl'], 1, 'input'); % Force applied by the NASS
io(4) = linio([mdl, '/Fdty'], 1, 'input'); % Parasitic force Ty
io(5) = linio([mdl, '/Fdrz'], 1, 'input'); % Parasitic force Rz
io(6) = linio([mdl, '/Dsm'], 1, 'output'); % Displacement of the sample
io(7) = linio([mdl, '/Fnlm'], 1, 'output'); % Force sensor in NASS's legs
io(8) = linio([mdl, '/Dnlm'], 1, 'output'); % Displacement of NASS's legs
io(9) = linio([mdl, '/Dgm'], 1, 'output'); % Absolute displacement of the granite
io(10) = linio([mdl, '/Vlm'], 1, 'output'); % Measured absolute velocity of the top NASS platform
#+end_src
#+begin_src matlab
%% Run the linearization
G_vc_iff = linearize(mdl, io, options);
G_vc_iff.InputName = {'Dw', ... % Ground Motion [m]
'Fs', ... % Force Applied on Sample [N]
'Fn', ... % Force applied by NASS [N]
'Fty', ... % Parasitic Force Ty [N]
'Frz'}; % Parasitic Force Rz [N]
G_vc_iff.OutputName = {'D', ... % Measured sample displacement x.r.t. granite [m]
'Fnm', ... % Force Sensor in NASS [N]
'Dnm', ... % Displacement Sensor in NASS [m]
'Dgm', ... % Asbolute displacement of Granite [m]
'Vlm'}; ... % Absolute Velocity of NASS [m/s]
#+end_src
** Noise Budget
We compute the obtain PSD of the displacement $D$ when using IFF.
#+begin_src matlab :exports none
% Power Spectral Density of the relative Displacement [m^2/Hz]
psd_vc_iff_gm_d = dist_f.psd_gm.*abs(squeeze(freqresp(G_vc_iff('D', 'Dw'), dist_f.f, 'Hz'))).^2;
psd_vc_iff_ty_d = dist_f.psd_ty.*abs(squeeze(freqresp(G_vc_iff('D', 'Fty'), dist_f.f, 'Hz'))).^2;
psd_vc_iff_rz_d = dist_f.psd_rz.*abs(squeeze(freqresp(G_vc_iff('D', 'Frz'), dist_f.f, 'Hz'))).^2;
#+end_src
#+begin_src matlab :exports none
freqs = logspace(0, 3, 1000);
figure;
hold on;
plot(dist_f.f, flip(sqrt(-cumtrapz(flip(dist_f.f), flip(psd_gm_d+psd_ty_d+psd_rz_d)))), '-', 'DisplayName', 'OL - PZ');
plot(dist_f.f, flip(sqrt(-cumtrapz(flip(dist_f.f), flip(psd_vc_gm_d+psd_vc_ty_d+psd_vc_rz_d)))), 'k-', 'DisplayName', 'OL - VC');
plot(dist_f.f, flip(sqrt(-cumtrapz(flip(dist_f.f), flip(psd_vc_iff_gm_d+psd_vc_iff_ty_d+psd_vc_iff_rz_d)))), 'k--', 'DisplayName', 'IFF - VC');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('CAS of the effect of disturbances on $D$ [m]'); xlabel('Frequency [Hz]');
legend('location', 'northeast')
xlim([0.5, 500]); ylim([1e-12, 5e-6]);
#+end_src
#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/uniaxial-cas-iff-vc.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+NAME: fig:uniaxial-cas-iff-vc
#+CAPTION: CAS of the displacement $D$ ([[./figs/uniaxial-cas-iff-vc.png][png]], [[./figs/uniaxial-cas-iff-vc.pdf][pdf]])
[[file:figs/uniaxial-cas-iff-vc.png]]
** Conclusion
#+begin_important
The use of voice coil actuators would allow a better disturbance rejection for a fixed bandwidth compared with a piezo-electric hexapod.
Similarly, it would require much lower bandwidth to attain the same level of disturbance rejection for $D$.
#+end_important

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