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< title > ESRF Double Crystal Monochromator - Dynamical Multi-Body Model< / title >
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< h1 class = "title" > ESRF Double Crystal Monochromator - Dynamical Multi-Body Model< / h1 >
< div id = "table-of-contents" role = "doc-toc" >
< h2 > Table of Contents< / h2 >
< div id = "text-table-of-contents" role = "doc-toc" >
< ul >
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< li > < a href = "#org7ab138f" > 1. Open Loop System Identification< / a >
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< ul >
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< li > < a href = "#org4bdd8a8" > 1.1. Identification< / a > < / li >
< li > < a href = "#org308c21c" > 1.2. Plant in the frame of the fastjacks< / a > < / li >
< li > < a href = "#org8722f96" > 1.3. Plant in the frame of the crystal< / a > < / li >
< / ul >
< / li >
< li > < a href = "#org7a5b9b3" > 2. Open-Loop Noise Budgeting< / a >
< ul >
< li > < a href = "#org331710d" > 2.1. Power Spectral Density of signals< / a > < / li >
< li > < a href = "#org3bca0d9" > 2.2. Open Loop disturbance and measurement noise< / a > < / li >
< / ul >
< / li >
< li > < a href = "#org808e143" > 3. Active Damping Plant (Strain gauges)< / a >
< ul >
< li > < a href = "#org0ab2a87" > 3.1. Identification< / a > < / li >
< li > < a href = "#org3834863" > 3.2. Relative Active Damping< / a > < / li >
< li > < a href = "#org972c8d8" > 3.3. Damped Plant< / a > < / li >
< / ul >
< / li >
< li > < a href = "#orgf406d83" > 4. Active Damping Plant (Force Sensors)< / a >
< ul >
< li > < a href = "#org9fe5c0e" > 4.1. Identification< / a > < / li >
< li > < a href = "#org3fa24b8" > 4.2. Controller - Root Locus< / a > < / li >
< li > < a href = "#orge5d3222" > 4.3. Damped Plant< / a > < / li >
< / ul >
< / li >
< li > < a href = "#org495a09b" > 5. Feedback Control< / a > < / li >
< li > < a href = "#orgefe0221" > 6. HAC-LAC (IFF) architecture< / a >
< ul >
< li > < a href = "#org1c6149b" > 6.1. System Identification< / a > < / li >
< li > < a href = "#orge6cba43" > 6.2. High Authority Controller< / a > < / li >
< li > < a href = "#org3cc0706" > 6.3. Performances< / a > < / li >
< li > < a href = "#org918f082" > 6.4. Close Loop noise budget< / a > < / li >
< / ul >
< / li >
< / ul >
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< / div >
< / div >
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< hr >
< p > This report is also available as a < a href = "./dcm-simscape-model.pdf" > pdf< / a > .< / p >
< hr >
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< p >
In this document, a Simscape (.e.g. multi-body) model of the ESRF Double Crystal Monochromator (DCM) is presented and used to develop and optimize the control strategy.
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< / p >
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< p >
It is structured as follow:
< / p >
< ul class = "org-ul" >
< li > Section < a href = "#org566f204" > 1< / a > : the system dynamics is identified in the absence of control.< / li >
< li > Section < a href = "#orgabe857c" > 2< / a > : an open-loop noise budget is performed.< / li >
< li > Section < a href = "#orgbcf21ac" > 3< / a > : it is studied whether if the strain gauges fixed to the piezoelectric actuators can be used to actively damp the plant.< / li >
< li > Section < a href = "#orgecb6c39" > 4< / a > : 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.< / li >
< li > Section < a href = "#org4ff1527" > 6< / a > : the High Authority Control - Low Authority Control (HAC-LAC) strategy is tested on the Simscape model.< / li >
< / ul >
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< div id = "outline-container-org7ab138f" class = "outline-2" >
< h2 id = "org7ab138f" > < span class = "section-number-2" > 1.< / span > Open Loop System Identification< / h2 >
< div class = "outline-text-2" id = "text-1" >
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< p >
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< a id = "org566f204" > < / a >
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< / p >
< / div >
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< div id = "outline-container-org4bdd8a8" class = "outline-3" >
< h3 id = "org4bdd8a8" > < span class = "section-number-3" > 1.1.< / span > Identification< / h3 >
< div class = "outline-text-3" id = "text-1-1" >
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< p >
Let’ s considered the system \(\bm{G}(s)\) with:
< / p >
< ul class = "org-ul" >
< li > 3 inputs: force applied to the 3 fast jacks< / li >
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< li > 3 outputs: measured displacement by the 3 interferometers pointing at the ring second crystal< / li >
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< / ul >
< p >
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It is schematically shown in Figure < a href = "#orgc25dd36" > 1< / a > .
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< / p >
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< div id = "orgc25dd36" class = "figure" >
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< p > < img src = "figs/schematic_system_inputs_outputs.png" alt = "schematic_system_inputs_outputs.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 1: < / span > Dynamical system with inputs and outputs< / p >
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< / div >
< p >
The system is identified from the Simscape model.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > %% Input/Output definition< / span >
clear io; io_i = 1;
< span class = "org-matlab-cellbreak" > %% Inputs< / span >
< span class = "org-comment-delimiter" > % < / span > < span class = "org-comment" > Control Input {3x1} [N]< / span >
io(io_i) = linio([mdl, < span class = "org-string" > '/control_system'< / span > ], 1, < span class = "org-string" > 'openinput'< / span > ); io_i = io_i < span class = "org-builtin" > +< / span > 1;
< span class = "org-matlab-cellbreak" > %% Outputs< / span >
< span class = "org-comment-delimiter" > % < / span > < span class = "org-comment" > Interferometers {3x1} [m]< / span >
io(io_i) = linio([mdl, < span class = "org-string" > '/DCM'< / span > ], 1, < span class = "org-string" > 'openoutput'< / span > ); io_i = io_i < span class = "org-builtin" > +< / span > 1;
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > %% Extraction of the dynamics< / span >
G = linearize(mdl, io);
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > size(G)
< / pre >
< / div >
< pre class = "example" >
size(G)
State-space model with 3 outputs, 3 inputs, and 24 states.
< / pre >
< / div >
< / div >
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< div id = "outline-container-org308c21c" class = "outline-3" >
< h3 id = "org308c21c" > < span class = "section-number-3" > 1.2.< / span > Plant in the frame of the fastjacks< / h3 >
< div class = "outline-text-3" id = "text-1-2" >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > load(< span class = "org-string" > 'dcm_kinematics.mat'< / span > );
< / pre >
< / div >
< p >
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Using the forward and inverse kinematics, we can computed the dynamics from piezo forces to axial motion of the 3 fastjacks (see Figure < a href = "#org3a46abb" > 2< / a > ).
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< / p >
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< div id = "org3a46abb" class = "figure" >
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< p > < img src = "figs/schematic_jacobian_frame_fastjack.png" alt = "schematic_jacobian_frame_fastjack.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 2: < / span > Use of Jacobian matrices to obtain the system in the frame of the fastjacks< / p >
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< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > %% Compute the system in the frame of the fastjacks< / span >
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G_pz = J_a_h< span class = "org-builtin" > *< / span > inv(J_2h_s)< span class = "org-builtin" > *< / span > G;
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< / pre >
< / div >
< p >
The DC gain of the new system shows that the system is well decoupled at low frequency.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > dcgain(G_pz)
< / pre >
< / div >
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< table id = "org16a1bc3" border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
< caption class = "t-above" > < span class = "table-number" > Table 1:< / span > DC gain of the plant in the frame of the fast jacks \(\bm{G}_{\text{fj}}\)< / caption >
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< colgroup >
< col class = "org-right" / >
< col class = "org-right" / >
< col class = "org-right" / >
< / colgroup >
< tbody >
< tr >
< td class = "org-right" > 4.4407e-09< / td >
< td class = "org-right" > 2.7656e-12< / td >
< td class = "org-right" > 1.0132e-12< / td >
< / tr >
< tr >
< td class = "org-right" > 2.7656e-12< / td >
< td class = "org-right" > 4.4407e-09< / td >
< td class = "org-right" > 1.0132e-12< / td >
< / tr >
< tr >
< td class = "org-right" > 1.0109e-12< / td >
< td class = "org-right" > 1.0109e-12< / td >
< td class = "org-right" > 4.4424e-09< / td >
< / tr >
< / tbody >
< / table >
< p >
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The bode plot of \(\bm{G}_{\text{fj}}(s)\) is shown in Figure < a href = "#orgd19fba2" > 3< / a > .
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< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > G_pz = diag(1< span class = "org-builtin" > ./< / span > diag(dcgain(G_pz)))< span class = "org-builtin" > *< / span > G_pz;
< / pre >
< / div >
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< div id = "orgd19fba2" class = "figure" >
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< p > < img src = "figs/bode_plot_plant_fj.png" alt = "bode_plot_plant_fj.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 3: < / span > Bode plot of the diagonal and off-diagonal elements of the plant in the frame of the fast jacks< / p >
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< / div >
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< div class = "important" id = "org70bf5d1" >
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< p >
Computing the system in the frame of the fastjack gives good decoupling at low frequency (until the first resonance of the system).
< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-org8722f96" class = "outline-3" >
< h3 id = "org8722f96" > < span class = "section-number-3" > 1.3.< / span > Plant in the frame of the crystal< / h3 >
< div class = "outline-text-3" id = "text-1-3" >
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< div id = "org331855a" class = "figure" >
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< p > < img src = "figs/schematic_jacobian_frame_crystal.png" alt = "schematic_jacobian_frame_crystal.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 4: < / span > Use of Jacobian matrices to obtain the system in the frame of the crystal< / p >
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< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > G_mr = inv(J_s_r)< span class = "org-builtin" > *< / span > G< span class = "org-builtin" > *< / span > inv(J_a_r< span class = "org-builtin" > '< / span > );
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< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > dcgain(G_mr)
< / pre >
< / div >
< table border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
< colgroup >
< col class = "org-right" / >
< col class = "org-right" / >
< col class = "org-right" / >
< / colgroup >
< tbody >
< tr >
< td class = "org-right" > 1.9978e-09< / td >
< td class = "org-right" > 3.9657e-09< / td >
< td class = "org-right" > 7.7944e-09< / td >
< / tr >
< tr >
< td class = "org-right" > 3.9656e-09< / td >
< td class = "org-right" > 8.4979e-08< / td >
< td class = "org-right" > -1.5135e-17< / td >
< / tr >
< tr >
< td class = "org-right" > 7.7944e-09< / td >
< td class = "org-right" > -3.9252e-17< / td >
< td class = "org-right" > 1.834e-07< / td >
< / tr >
< / tbody >
< / table >
< p >
This results in a coupled system.
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The main reason is that, as we map forces to the center of the ring crystal and not at the center of mass/stiffness of the moving part, vertical forces will induce rotation and torques will induce vertical motion.
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< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-org7a5b9b3" class = "outline-2" >
< h2 id = "org7a5b9b3" > < span class = "section-number-2" > 2.< / span > Open-Loop Noise Budgeting< / h2 >
< div class = "outline-text-2" id = "text-2" >
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< p >
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< a id = "orgabe857c" > < / a >
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< / p >
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< div id = "orgef00456" class = "figure" >
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< p > < img src = "figs/noise_budget_dcm_schematic_fast_jack_frame.png" alt = "noise_budget_dcm_schematic_fast_jack_frame.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 5: < / span > Schematic representation of the control loop in the frame of one fast jack< / p >
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< / div >
< / div >
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< div id = "outline-container-org331710d" class = "outline-3" >
< h3 id = "org331710d" > < span class = "section-number-3" > 2.1.< / span > Power Spectral Density of signals< / h3 >
< div class = "outline-text-3" id = "text-2-1" >
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< p >
Interferometer noise:
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Wn = 6e< span class = "org-builtin" > -< / span > 11< span class = "org-builtin" > *< / span > (1 < span class = "org-builtin" > +< / span > s< span class = "org-builtin" > /< / span > 2< span class = "org-builtin" > /< / span > < span class = "org-matlab-math" > pi< / span > < span class = "org-builtin" > /< / span > 200)< span class = "org-builtin" > /< / span > (1 < span class = "org-builtin" > +< / span > s< span class = "org-builtin" > /< / span > 2< span class = "org-builtin" > /< / span > < span class = "org-matlab-math" > pi< / span > < span class = "org-builtin" > /< / span > 60); < span class = "org-comment-delimiter" > % < / span > < span class = "org-comment" > m/sqrt(Hz)< / span >
< / pre >
< / div >
< pre class = "example" >
Measurement noise: 0.79 [nm,rms]
< / pre >
< p >
DAC noise (amplified by the PI voltage amplifier, and converted to newtons):
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Wdac = tf(3e< span class = "org-builtin" > -< / span > 8); < span class = "org-comment-delimiter" > % < / span > < span class = "org-comment" > V/sqrt(Hz)< / span >
Wu = Wdac< span class = "org-builtin" > *< / span > 22.5< span class = "org-builtin" > *< / span > 10; < span class = "org-comment-delimiter" > % < / span > < span class = "org-comment" > N/sqrt(Hz)< / span >
< / pre >
< / div >
< pre class = "example" >
DAC noise: 0.95 [uV,rms]
< / pre >
< p >
Disturbances:
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Wd = 5e< span class = "org-builtin" > -< / span > 7< span class = "org-builtin" > /< / span > (1 < span class = "org-builtin" > +< / span > s< span class = "org-builtin" > /< / span > 2< span class = "org-builtin" > /< / span > < span class = "org-matlab-math" > pi< / span > ); < span class = "org-comment-delimiter" > % < / span > < span class = "org-comment" > m/sqrt(Hz)< / span >
< / pre >
< / div >
< pre class = "example" >
Disturbance motion: 0.61 [um,rms]
< / pre >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > %% Save ASD of noise and disturbances< / span >
save(< span class = "org-string" > 'mat/asd_noises_disturbances.mat'< / span > , < span class = "org-string" > 'Wn'< / span > , < span class = "org-string" > 'Wu'< / span > , < span class = "org-string" > 'Wd'< / span > );
< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org3bca0d9" class = "outline-3" >
< h3 id = "org3bca0d9" > < span class = "section-number-3" > 2.2.< / span > Open Loop disturbance and measurement noise< / h3 >
< div class = "outline-text-3" id = "text-2-2" >
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< p >
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The comparison of the amplitude spectral density of the measurement noise and of the jack parasitic motion is performed in Figure < a href = "#orgf0b0f03" > 6< / a > .
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It confirms that the sensor noise is low enough to measure the motion errors of the crystal.
< / p >
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< div id = "orgf0b0f03" class = "figure" >
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< p > < img src = "figs/open_loop_noise_budget_fast_jack.png" alt = "open_loop_noise_budget_fast_jack.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 6: < / span > Open Loop noise budgeting< / p >
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< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-org808e143" class = "outline-2" >
< h2 id = "org808e143" > < span class = "section-number-2" > 3.< / span > Active Damping Plant (Strain gauges)< / h2 >
< div class = "outline-text-2" id = "text-3" >
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< p >
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< a id = "orgbcf21ac" > < / a >
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< / p >
< p >
In this section, we wish to see whether if strain gauges fixed to the piezoelectric actuator can be used for active damping.
< / p >
< / div >
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< div id = "outline-container-org0ab2a87" class = "outline-3" >
< h3 id = "org0ab2a87" > < span class = "section-number-3" > 3.1.< / span > Identification< / h3 >
< div class = "outline-text-3" id = "text-3-1" >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > %% Input/Output definition< / span >
clear io; io_i = 1;
< span class = "org-matlab-cellbreak" > %% Inputs< / span >
< span class = "org-comment-delimiter" > % < / span > < span class = "org-comment" > Control Input {3x1} [N]< / span >
io(io_i) = linio([mdl, < span class = "org-string" > '/control_system'< / span > ], 1, < span class = "org-string" > 'openinput'< / span > ); io_i = io_i < span class = "org-builtin" > +< / span > 1;
< span class = "org-matlab-cellbreak" > %% Outputs< / span >
< span class = "org-comment-delimiter" > % < / span > < span class = "org-comment" > Strain Gauges {3x1} [m]< / span >
io(io_i) = linio([mdl, < span class = "org-string" > '/DCM'< / span > ], 2, < span class = "org-string" > 'openoutput'< / span > ); io_i = io_i < span class = "org-builtin" > +< / span > 1;
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > %% Extraction of the dynamics< / span >
G_sg = linearize(mdl, io);
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > dcgain(G_sg)
< / pre >
< / div >
< table border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
< colgroup >
< col class = "org-right" / >
< col class = "org-right" / >
< col class = "org-right" / >
< / colgroup >
< tbody >
< tr >
< td class = "org-right" > 4.4443e-09< / td >
< td class = "org-right" > 1.0339e-13< / td >
< td class = "org-right" > 3.774e-14< / td >
< / tr >
< tr >
< td class = "org-right" > 1.0339e-13< / td >
< td class = "org-right" > 4.4443e-09< / td >
< td class = "org-right" > 3.774e-14< / td >
< / tr >
< tr >
< td class = "org-right" > 3.7792e-14< / td >
< td class = "org-right" > 3.7792e-14< / td >
< td class = "org-right" > 4.4444e-09< / td >
< / tr >
< / tbody >
< / table >
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< div id = "orgfdda5de" class = "figure" >
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< p > < img src = "figs/strain_gauge_plant_bode_plot.png" alt = "strain_gauge_plant_bode_plot.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 7: < / span > Bode Plot of the transfer functions from piezoelectric forces to strain gauges measuremed displacements< / p >
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< / div >
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< div class = "important" id = "org8c0fd92" >
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< p >
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As the distance between the poles and zeros in Figure < a href = "#orgd1f744c" > 10< / a > is very small, little damping can be actively added using the strain gauges.
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This will be confirmed using a Root Locus plot.
< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-org3834863" class = "outline-3" >
< h3 id = "org3834863" > < span class = "section-number-3" > 3.2.< / span > Relative Active Damping< / h3 >
< div class = "outline-text-3" id = "text-3-2" >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > Krad_g1 = eye(3)< span class = "org-builtin" > *< / span > s< span class = "org-builtin" > /< / span > (s< span class = "org-builtin" > ^< / span > 2< span class = "org-builtin" > /< / span > (2< span class = "org-builtin" > *< / span > < span class = "org-matlab-math" > pi< / span > < span class = "org-builtin" > *< / span > 500)< span class = "org-builtin" > ^< / span > 2 < span class = "org-builtin" > +< / span > 2< span class = "org-builtin" > *< / span > s< span class = "org-builtin" > /< / span > (2< span class = "org-builtin" > *< / span > < span class = "org-matlab-math" > pi< / span > < span class = "org-builtin" > *< / span > 500) < span class = "org-builtin" > +< / span > 1);
< / pre >
< / div >
< p >
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As can be seen in Figure < a href = "#orgd1eb86d" > 8< / a > , very little damping can be added using relative damping strategy using strain gauges.
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< / p >
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< div id = "orgd1eb86d" class = "figure" >
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< p > < img src = "figs/relative_damping_root_locus.png" alt = "relative_damping_root_locus.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 8: < / span > Root Locus for the relative damping control< / p >
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< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Krad = < span class = "org-builtin" > -< / span > g< span class = "org-builtin" > *< / span > Krad_g1;
< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org972c8d8" class = "outline-3" >
< h3 id = "org972c8d8" > < span class = "section-number-3" > 3.3.< / span > Damped Plant< / h3 >
< div class = "outline-text-3" id = "text-3-3" >
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< p >
The controller is implemented on Simscape, and the damped plant is identified.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > %% Input/Output definition< / span >
clear io; io_i = 1;
< span class = "org-matlab-cellbreak" > %% Inputs< / span >
< span class = "org-comment-delimiter" > % < / span > < span class = "org-comment" > Control Input {3x1} [N]< / span >
io(io_i) = linio([mdl, < span class = "org-string" > '/control_system'< / span > ], 1, < span class = "org-string" > 'input'< / span > ); io_i = io_i < span class = "org-builtin" > +< / span > 1;
< span class = "org-matlab-cellbreak" > %% Outputs< / span >
< span class = "org-comment-delimiter" > % < / span > < span class = "org-comment" > Force Sensor {3x1} [m]< / span >
io(io_i) = linio([mdl, < span class = "org-string" > '/DCM'< / span > ], 1, < span class = "org-string" > 'openoutput'< / span > ); io_i = io_i < span class = "org-builtin" > +< / span > 1;
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > %% DCM Kinematics< / span >
load(< span class = "org-string" > 'dcm_kinematics.mat'< / span > );
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > %% Identification of the Open Loop plant< / span >
controller.type = 0; < span class = "org-comment-delimiter" > % < / span > < span class = "org-comment" > Open Loop< / span >
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G_ol = J_a_r< span class = "org-builtin" > *< / span > inv(J_s_r)< span class = "org-builtin" > *< / span > linearize(mdl, io);
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G_ol.InputName = {< span class = "org-string" > 'u_ur'< / span > , < span class = "org-string" > 'u_uh'< / span > , < span class = "org-string" > 'u_d'< / span > };
G_ol.OutputName = {< span class = "org-string" > 'd_ur'< / span > , < span class = "org-string" > 'd_uh'< / span > , < span class = "org-string" > 'd_d'< / span > };
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > %% Identification of the damped plant with Relative Active Damping< / span >
controller.type = 2; < span class = "org-comment-delimiter" > % < / span > < span class = "org-comment" > RAD< / span >
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G_dp = J_a_r< span class = "org-builtin" > *< / span > inv(J_s_r)< span class = "org-builtin" > *< / span > linearize(mdl, io);
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G_dp.InputName = {< span class = "org-string" > 'u_ur'< / span > , < span class = "org-string" > 'u_uh'< / span > , < span class = "org-string" > 'u_d'< / span > };
G_dp.OutputName = {< span class = "org-string" > 'd_ur'< / span > , < span class = "org-string" > 'd_uh'< / span > , < span class = "org-string" > 'd_d'< / span > };
< / pre >
< / div >
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< div id = "orgdfa8a77" class = "figure" >
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< p > < img src = "figs/comp_damp_undamped_plant_rad_bode_plot.png" alt = "comp_damp_undamped_plant_rad_bode_plot.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 9: < / span > Bode plot of both the open-loop plant and the damped plant using relative active damping< / p >
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< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgf406d83" class = "outline-2" >
< h2 id = "orgf406d83" > < span class = "section-number-2" > 4.< / span > Active Damping Plant (Force Sensors)< / h2 >
< div class = "outline-text-2" id = "text-4" >
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< p >
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< a id = "orgecb6c39" > < / a >
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< / p >
< p >
Force sensors are added above the piezoelectric actuators.
They can consists of a simple piezoelectric ceramic stack.
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See for instance (< a href = "#citeproc_bib_item_1" > Fleming and Leang 2010< / a > ).
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< / p >
< / div >
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< div id = "outline-container-org9fe5c0e" class = "outline-3" >
< h3 id = "org9fe5c0e" > < span class = "section-number-3" > 4.1.< / span > Identification< / h3 >
< div class = "outline-text-3" id = "text-4-1" >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > %% Input/Output definition< / span >
clear io; io_i = 1;
< span class = "org-matlab-cellbreak" > %% Inputs< / span >
< span class = "org-comment-delimiter" > % < / span > < span class = "org-comment" > Control Input {3x1} [N]< / span >
io(io_i) = linio([mdl, < span class = "org-string" > '/control_system'< / span > ], 1, < span class = "org-string" > 'openinput'< / span > ); io_i = io_i < span class = "org-builtin" > +< / span > 1;
< span class = "org-matlab-cellbreak" > %% Outputs< / span >
< span class = "org-comment-delimiter" > % < / span > < span class = "org-comment" > Force Sensor {3x1} [m]< / span >
io(io_i) = linio([mdl, < span class = "org-string" > '/DCM'< / span > ], 3, < span class = "org-string" > 'openoutput'< / span > ); io_i = io_i < span class = "org-builtin" > +< / span > 1;
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > %% Extraction of the dynamics< / span >
G_fs = linearize(mdl, io);
< / pre >
< / div >
< p >
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The Bode plot of the identified dynamics is shown in Figure < a href = "#orgd1f744c" > 10< / a > .
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At high frequency, the diagonal terms are constants while the off-diagonal terms have some roll-off.
< / p >
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< div id = "orgd1f744c" class = "figure" >
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< p > < img src = "figs/iff_plant_bode_plot.png" alt = "iff_plant_bode_plot.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 10: < / span > Bode plot of IFF Plant< / p >
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< / div >
< / div >
< / div >
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< div id = "outline-container-org3fa24b8" class = "outline-3" >
< h3 id = "org3fa24b8" > < span class = "section-number-3" > 4.2.< / span > Controller - Root Locus< / h3 >
< div class = "outline-text-3" id = "text-4-2" >
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< p >
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.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > %% Integral Force Feedback Controller< / span >
Kiff_g1 = eye(3)< span class = "org-builtin" > *< / span > 1< span class = "org-builtin" > /< / span > (1 < span class = "org-builtin" > +< / span > s< span class = "org-builtin" > /< / span > 2< span class = "org-builtin" > /< / span > < span class = "org-matlab-math" > pi< / span > < span class = "org-builtin" > /< / span > 20);
< / pre >
< / div >
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< div id = "org7f81ce5" class = "figure" >
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< p > < img src = "figs/iff_root_locus.png" alt = "iff_root_locus.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 11: < / span > Root Locus plot for the IFF Control strategy< / p >
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< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > %% Integral Force Feedback Controller with optimal gain< / span >
Kiff = g< span class = "org-builtin" > *< / span > Kiff_g1;
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > %% Save the IFF controller< / span >
save(< span class = "org-string" > 'mat/Kiff.mat'< / span > , < span class = "org-string" > 'Kiff'< / span > );
< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-orge5d3222" class = "outline-3" >
< h3 id = "orge5d3222" > < span class = "section-number-3" > 4.3.< / span > Damped Plant< / h3 >
< div class = "outline-text-3" id = "text-4-3" >
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< p >
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Both the Open Loop dynamics (see Figure < a href = "#org3a46abb" > 2< / a > ) and the dynamics with IFF (see Figure < a href = "#org4aafce8" > 12< / a > ) are identified.
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< / p >
< p >
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).
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This is schematically represented in Figure < a href = "#org4aafce8" > 12< / a > .
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< / p >
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< div id = "org4aafce8" class = "figure" >
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< p > < img src = "figs/schematic_jacobian_frame_fastjack_iff.png" alt = "schematic_jacobian_frame_fastjack_iff.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 12: < / span > Use of Jacobian matrices to obtain the system in the frame of the fastjacks< / p >
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< / div >
< p >
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The dynamics from \(\bm{u}\) to \(\bm{d}_{\text{fj}}\) (open-loop dynamics) and from \(\bm{u}^\prime\) to \(\bm{d}_{\text{fs}}\) are compared in Figure < a href = "#orgbd8afb9" > 13< / a > .
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It is clear that the Integral Force Feedback control strategy is very effective in damping the resonances of the plant.
< / p >
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< div id = "orgbd8afb9" class = "figure" >
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< p > < img src = "figs/comp_damped_undamped_plant_iff_bode_plot.png" alt = "comp_damped_undamped_plant_iff_bode_plot.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 13: < / span > Bode plot of both the open-loop plant and the damped plant using IFF< / p >
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< / div >
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< div class = "important" id = "org53702ee" >
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< p >
The Integral Force Feedback control strategy is very effective in damping the modes present in the plant.
< / p >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-org495a09b" class = "outline-2" >
< h2 id = "org495a09b" > < span class = "section-number-2" > 5.< / span > Feedback Control< / h2 >
< div class = "outline-text-2" id = "text-5" >
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< div id = "org14950fe" class = "figure" >
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< p > < img src = "figs/schematic_jacobian_frame_fastjack_feedback.png" alt = "schematic_jacobian_frame_fastjack_feedback.png" / >
< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgefe0221" class = "outline-2" >
< h2 id = "orgefe0221" > < span class = "section-number-2" > 6.< / span > HAC-LAC (IFF) architecture< / h2 >
< div class = "outline-text-2" id = "text-6" >
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< p >
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< a id = "org4ff1527" > < / a >
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< / p >
< p >
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The HAC-LAC architecture is shown in Figure < a href = "#org8909d7d" > 14< / a > .
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< / p >
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< div id = "org8909d7d" class = "figure" >
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< p > < img src = "figs/schematic_jacobian_frame_fastjack_hac_iff.png" alt = "schematic_jacobian_frame_fastjack_hac_iff.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 14: < / span > HAC-LAC architecture< / p >
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< / div >
< / div >
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< div id = "outline-container-org1c6149b" class = "outline-3" >
< h3 id = "org1c6149b" > < span class = "section-number-3" > 6.1.< / span > System Identification< / h3 >
< div class = "outline-text-3" id = "text-6-1" >
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< p >
Let’ s identify the damped plant.
< / p >
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< div id = "org1ad0ce7" class = "figure" >
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< p > < img src = "figs/bode_plot_hac_iff_plant.png" alt = "bode_plot_hac_iff_plant.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 15: < / span > Bode Plot of the plant for the High Authority Controller (transfer function from \(\bm{u}^\prime\) to \(\bm{\epsilon}_d\))< / p >
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< / div >
< / div >
< / div >
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< div id = "outline-container-orge6cba43" class = "outline-3" >
< h3 id = "orge6cba43" > < span class = "section-number-3" > 6.2.< / span > High Authority Controller< / h3 >
< div class = "outline-text-3" id = "text-6-2" >
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< p >
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.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > %% Controller design< / span >
wc = 2< span class = "org-builtin" > *< / span > < span class = "org-matlab-math" > pi< / span > < span class = "org-builtin" > *< / span > 100; < span class = "org-comment-delimiter" > % < / span > < span class = "org-comment" > Wanted crossover frequency [rad/s]< / span >
a = 2; < span class = "org-comment-delimiter" > % < / span > < span class = "org-comment" > Lead parameter< / span >
Khac = diag(1< span class = "org-builtin" > ./< / span > diag(abs(evalfr(G_dp, 1< span class = "org-constant" > j< / span > < span class = "org-builtin" > *< / span > wc)))) < span class = "org-builtin" > *< / span > < span class = "org-comment-delimiter" > .< / span > < span class = "org-comment" > .. % Diagonal controller< / span >
wc< span class = "org-builtin" > /< / span > s < span class = "org-builtin" > *< / span > < span class = "org-comment-delimiter" > .< / span > < span class = "org-comment" > .. % Integrator< / span >
1< span class = "org-builtin" > /< / span > (sqrt(a))< span class = "org-builtin" > *< / span > (1 < span class = "org-builtin" > +< / span > s< span class = "org-builtin" > /< / span > (wc< span class = "org-builtin" > /< / span > sqrt(a)))< span class = "org-builtin" > /< / span > (1 < span class = "org-builtin" > +< / span > s< span class = "org-builtin" > /< / span > (wc< span class = "org-builtin" > *< / span > sqrt(a))) < span class = "org-builtin" > *< / span > < span class = "org-comment-delimiter" > .< / span > < span class = "org-comment" > .. % Lead< / span >
1< span class = "org-builtin" > /< / span > (s< span class = "org-builtin" > ^< / span > 2< span class = "org-builtin" > /< / span > (4< span class = "org-builtin" > *< / span > wc)< span class = "org-builtin" > ^< / span > 2 < span class = "org-builtin" > +< / span > 2< span class = "org-builtin" > *< / span > s< span class = "org-builtin" > /< / span > (4< span class = "org-builtin" > *< / span > wc) < span class = "org-builtin" > +< / span > 1); < span class = "org-comment-delimiter" > % < / span > < span class = "org-comment" > Low pass filter< / span >
< / pre >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > %% Save the HAC controller< / span >
save(< span class = "org-string" > 'mat/Khac_iff.mat'< / span > , < span class = "org-string" > 'Khac'< / span > );
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > %% Loop Gain< / span >
L_hac_lac = G_dp < span class = "org-builtin" > *< / span > Khac;
< / pre >
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< div id = "orgeac5939" class = "figure" >
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< p > < img src = "figs/hac_iff_loop_gain_bode_plot.png" alt = "hac_iff_loop_gain_bode_plot.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 16: < / span > Bode Plot of the Loop gain for the High Authority Controller< / p >
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< / div >
< p >
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As shown in the Root Locus plot in Figure < a href = "#org48f5645" > 17< / a > , the closed loop system should be stable.
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< / p >
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< div id = "org48f5645" class = "figure" >
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< p > < img src = "figs/loci_hac_iff_fast_jack.png" alt = "loci_hac_iff_fast_jack.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 17: < / span > Root Locus for the High Authority Controller< / p >
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< / div >
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< div id = "outline-container-org3cc0706" class = "outline-3" >
< h3 id = "org3cc0706" > < span class = "section-number-3" > 6.3.< / span > Performances< / h3 >
< div class = "outline-text-3" id = "text-6-3" >
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< p >
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.
< / p >
< p >
It is first verified that the closed-loop system is stable:
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > isstable(T_hl)
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< pre class = "example" >
1
< / pre >
< p >
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And both transmissibilities are compared in Figure < a href = "#org9714ce1" > 18< / a > .
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< / p >
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< div id = "org9714ce1" class = "figure" >
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< p > < img src = "figs/stepper_transmissibility_comp_ol_hac_iff.png" alt = "stepper_transmissibility_comp_ol_hac_iff.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 18: < / span > Comparison of the transmissibility of errors from vibrations of the stepper motor between the open-loop case and the hac-iff case.< / p >
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< / div >
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< div class = "important" id = "org9958f45" >
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< p >
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.
< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-org918f082" class = "outline-3" >
< h3 id = "org918f082" > < span class = "section-number-3" > 6.4.< / span > Close Loop noise budget< / h3 >
< div class = "outline-text-3" id = "text-6-4" >
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< p >
Let’ s compute the amplitude spectral density of the jack motion errors due to the sensor noise, the actuator noise and disturbances.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > %% Computation of ASD of contribution of inputs to the closed-loop motion< / span >
< span class = "org-comment-delimiter" > % < / span > < span class = "org-comment" > Error due to disturbances< / span >
asd_d = abs(squeeze(freqresp(Wd< span class = "org-builtin" > *< / span > (1< span class = "org-builtin" > /< / span > (1 < span class = "org-builtin" > +< / span > G_dp(1,1)< span class = "org-builtin" > *< / span > Khac(1,1))), f, < span class = "org-string" > 'Hz'< / span > )));
< span class = "org-comment-delimiter" > % < / span > < span class = "org-comment" > Error due to actuator noise< / span >
asd_u = abs(squeeze(freqresp(Wu< span class = "org-builtin" > *< / span > (G_dp(1,1)< span class = "org-builtin" > /< / span > (1 < span class = "org-builtin" > +< / span > G_dp(1,1)< span class = "org-builtin" > *< / span > Khac(1,1))), f, < span class = "org-string" > 'Hz'< / span > )));
< span class = "org-comment-delimiter" > % < / span > < span class = "org-comment" > Error due to sensor noise< / span >
asd_n = abs(squeeze(freqresp(Wn< span class = "org-builtin" > *< / span > (G_dp(1,1)< span class = "org-builtin" > *< / span > Khac(1,1)< span class = "org-builtin" > /< / span > (1 < span class = "org-builtin" > +< / span > G_dp(1,1)< span class = "org-builtin" > *< / span > Khac(1,1))), f, < span class = "org-string" > 'Hz'< / span > )));
< / pre >
< / div >
< p >
The closed-loop ASD is then:
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > %% ASD of the closed-loop motion< / span >
asd_cl = sqrt(asd_d< span class = "org-builtin" > .^< / span > 2 < span class = "org-builtin" > +< / span > asd_u< span class = "org-builtin" > .^< / span > 2 < span class = "org-builtin" > +< / span > asd_n< span class = "org-builtin" > .^< / span > 2);
< / pre >
< / div >
< p >
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The obtained ASD are shown in Figure < a href = "#org0490d21" > 19< / a > .
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< / p >
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< div id = "org0490d21" class = "figure" >
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< p > < img src = "figs/close_loop_asd_noise_budget_hac_iff.png" alt = "close_loop_asd_noise_budget_hac_iff.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 19: < / span > Closed Loop noise budget< / p >
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< / div >
< p >
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Let’ s compare the open-loop and close-loop cases (Figure < a href = "#orgcda44ba" > 20< / a > ).
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< / p >
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< div id = "orgcda44ba" class = "figure" >
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< p > < img src = "figs/cps_comp_ol_cl_hac_iff.png" alt = "cps_comp_ol_cl_hac_iff.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 20: < / span > Cumulative Power Spectrum of the open-loop and closed-loop motion error along one fast-jack< / p >
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< / div >
< / div >
< / div >
< / div >
< / div >
< div id = "postamble" class = "status" >
< p class = "author" > Author: Dehaeze Thomas< / p >
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< p class = "date" > Created: 2022-06-02 Thu 18:11< / p >
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