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< title > HAC-LAC applied on the Simscape Model< / title >
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< meta name = "author" content = "Dehaeze Thomas" / >
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< / div > < div id = "content" >
< h1 class = "title" > HAC-LAC applied on the Simscape Model< / h1 >
< div id = "table-of-contents" >
< h2 > Table of Contents< / h2 >
< div id = "text-table-of-contents" >
< ul >
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< li > < a href = "#org689cb19" > 1. Initialization< / a > < / li >
< li > < a href = "#org24d1d7f" > 2. Low Authority Control - Direct Velocity Feedback \(\bm{K}_\mathcal{L}\)< / a >
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< ul >
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< li > < a href = "#org68cbbe7" > 2.1. Identification< / a > < / li >
< li > < a href = "#org7c1c629" > 2.2. Plant< / a > < / li >
< li > < a href = "#org719e252" > 2.3. Root Locus< / a > < / li >
< li > < a href = "#org2c42b90" > 2.4. Controller and Loop Gain< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org195a29c" > 3. Uncertainty Improvements thanks to the LAC control< / a > < / li >
< li > < a href = "#org81738ac" > 4. High Authority Control - \(\bm{K}_\mathcal{X}\)< / a >
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< ul >
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< li > < a href = "#org94cb2c9" > 4.1. Identification of the damped plant< / a > < / li >
< li > < a href = "#org2a269d5" > 4.2. Controller Design< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org1171960" > 5. Simulation< / a > < / li >
< li > < a href = "#orgad3e1c7" > 6. Results< / a > < / li >
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< / ul >
< / div >
< / div >
< p >
The position \(\bm{\mathcal{X}}\) of the Sample with respect to the granite is measured.
< / p >
< p >
It is then compare to the wanted position of the Sample \(\bm{r}_\mathcal{X}\) in order to obtain the position error \(\bm{\epsilon}_\mathcal{X}\) of the Sample with respect to a frame attached to the Stewart top platform.
< / p >
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< div id = "org270f668" class = "figure" >
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< p > < img src = "figs/hac_lac_control_schematic.png" alt = "hac_lac_control_schematic.png" / >
< / p >
< p > < span class = "figure-number" > Figure 1: < / span > HAC-LAC Control Architecture used for the Control of the NASS< / p >
< / div >
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< div id = "outline-container-org689cb19" class = "outline-2" >
< h2 id = "org689cb19" > < span class = "section-number-2" > 1< / span > Initialization< / h2 >
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< div class = "outline-text-2" id = "text-1" >
< p >
We initialize all the stages with the default parameters.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeAxisc();
initializeMirror();
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< / pre >
< / div >
< p >
The nano-hexapod is a piezoelectric hexapod and the sample has a mass of 50kg.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > initializeNanoHexapod(< span class = "org-string" > 'actuator'< / span > , < span class = "org-string" > 'piezo'< / span > );
initializeSample(< span class = "org-string" > 'mass'< / span > , 1);
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< / pre >
< / div >
< p >
We set the references that corresponds to a tomography experiment.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > initializeReferences(< span class = "org-string" > 'Rz_type'< / span > , < span class = "org-string" > 'rotating'< / span > , < span class = "org-string" > 'Rz_period'< / span > , 1);
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > initializeDisturbances();
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< / pre >
< / div >
< p >
Open Loop.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > initializeController(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'open-loop'< / span > );
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< / pre >
< / div >
< p >
And we put some gravity.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > initializeSimscapeConfiguration(< span class = "org-string" > 'gravity'< / span > , < span class = "org-constant" > true< / span > );
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< / pre >
< / div >
< p >
We log the signals.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > initializeLoggingConfiguration(< span class = "org-string" > 'log'< / span > , < span class = "org-string" > 'all'< / span > );
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< / pre >
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< div id = "outline-container-org24d1d7f" class = "outline-2" >
< h2 id = "org24d1d7f" > < span class = "section-number-2" > 2< / span > Low Authority Control - Direct Velocity Feedback \(\bm{K}_\mathcal{L}\)< / h2 >
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< div class = "outline-text-2" id = "text-2" >
< p >
The first loop closed corresponds to a direct velocity feedback loop.
< / p >
< p >
The design of the associated decentralized controller is explained in < a href = "control_active_damping.html" > this< / a > file.
< / p >
< / div >
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< div id = "outline-container-org68cbbe7" class = "outline-3" >
< h3 id = "org68cbbe7" > < span class = "section-number-3" > 2.1< / span > Identification< / h3 >
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< div class = "outline-text-3" id = "text-2-1" >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Name of the Simulink File< / span > < / span >
mdl = < span class = "org-string" > 'nass_model'< / span > ;
< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Input/Output definition< / span > < / span >
clear io; io_i = 1;
io(io_i) = linio([mdl, < span class = "org-string" > '/Controller'< / span > ], 1, < span class = "org-string" > 'openinput'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Actuator Inputs< / span >
io(io_i) = linio([mdl, < span class = "org-string" > '/Micro-Station'< / span > ], 3, < span class = "org-string" > 'openoutput'< / span > , [], < span class = "org-string" > 'Dnlm'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Relative Motion Outputs< / span >
< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Run the linearization< / span > < / span >
G_dvf = linearize(mdl, io, 0);
G_dvf.InputName = {< span class = "org-string" > 'Fnl1'< / span > , < span class = "org-string" > 'Fnl2'< / span > , < span class = "org-string" > 'Fnl3'< / span > , < span class = "org-string" > 'Fnl4'< / span > , < span class = "org-string" > 'Fnl5'< / span > , < span class = "org-string" > 'Fnl6'< / span > };
G_dvf.OutputName = {< span class = "org-string" > 'Dnlm1'< / span > , < span class = "org-string" > 'Dnlm2'< / span > , < span class = "org-string" > 'Dnlm3'< / span > , < span class = "org-string" > 'Dnlm4'< / span > , < span class = "org-string" > 'Dnlm5'< / span > , < span class = "org-string" > 'Dnlm6'< / span > };
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< / pre >
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< / div >
< / div >
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< div id = "outline-container-org7c1c629" class = "outline-3" >
< h3 id = "org7c1c629" > < span class = "section-number-3" > 2.2< / span > Plant< / h3 >
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< / div >
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< div id = "outline-container-org719e252" class = "outline-3" >
< h3 id = "org719e252" > < span class = "section-number-3" > 2.3< / span > Root Locus< / h3 >
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< / div >
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< div id = "outline-container-org2c42b90" class = "outline-3" >
< h3 id = "org2c42b90" > < span class = "section-number-3" > 2.4< / span > Controller and Loop Gain< / h3 >
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< div class = "outline-text-3" id = "text-2-4" >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > K_dvf = s< span class = "org-type" > *< / span > 15000< span class = "org-type" > /< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > 2< span class = "org-type" > /< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > /< / span > 10000);
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > K_dvf = < span class = "org-type" > -< / span > K_dvf< span class = "org-type" > *< / span > eye(6);
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< / pre >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-org195a29c" class = "outline-2" >
< h2 id = "org195a29c" > < span class = "section-number-2" > 3< / span > Uncertainty Improvements thanks to the LAC control< / h2 >
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< div class = "outline-text-2" id = "text-3" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > K_dvf_backup = K_dvf;
initializeController(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'hac-dvf'< / span > );
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > masses = [1, 10, 50]; < span class = "org-comment" > % [kg]< / span >
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Name of the Simulink File< / span > < / span >
mdl = < span class = "org-string" > 'nass_model'< / span > ;
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< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Input/Output definition< / span > < / span >
clear io; io_i = 1;
io(io_i) = linio([mdl, < span class = "org-string" > '/Controller'< / span > ], 1, < span class = "org-string" > 'input'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Actuator Inputs< / span >
io(io_i) = linio([mdl, < span class = "org-string" > '/Tracking Error'< / span > ], 1, < span class = "org-string" > 'output'< / span > , [], < span class = "org-string" > 'En'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Position Errror< / span >
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< / pre >
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< / div >
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< div id = "outline-container-org81738ac" class = "outline-2" >
< h2 id = "org81738ac" > < span class = "section-number-2" > 4< / span > High Authority Control - \(\bm{K}_\mathcal{X}\)< / h2 >
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< div class = "outline-text-2" id = "text-4" >
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< / div >
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< div id = "outline-container-org94cb2c9" class = "outline-3" >
< h3 id = "org94cb2c9" > < span class = "section-number-3" > 4.1< / span > Identification of the damped plant< / h3 >
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< div class = "outline-text-3" id = "text-4-1" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > Kx = tf(zeros(6));
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > initializeController(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'hac-dvf'< / span > );
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Name of the Simulink File< / span > < / span >
mdl = < span class = "org-string" > 'nass_model'< / span > ;
< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Input/Output definition< / span > < / span >
clear io; io_i = 1;
io(io_i) = linio([mdl, < span class = "org-string" > '/Controller'< / span > ], 1, < span class = "org-string" > 'input'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Actuator Inputs< / span >
io(io_i) = linio([mdl, < span class = "org-string" > '/Tracking Error'< / span > ], 1, < span class = "org-string" > 'output'< / span > , [], < span class = "org-string" > 'En'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Position Errror< / span >
< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Run the linearization< / span > < / span >
G = linearize(mdl, io, 0);
G.InputName = {< span class = "org-string" > 'Fnl1'< / span > , < span class = "org-string" > 'Fnl2'< / span > , < span class = "org-string" > 'Fnl3'< / span > , < span class = "org-string" > 'Fnl4'< / span > , < span class = "org-string" > 'Fnl5'< / span > , < span class = "org-string" > 'Fnl6'< / span > };
G.OutputName = {< span class = "org-string" > 'Ex'< / span > , < span class = "org-string" > 'Ey'< / span > , < span class = "org-string" > 'Ez'< / span > , < span class = "org-string" > 'Erx'< / span > , < span class = "org-string" > 'Ery'< / span > , < span class = "org-string" > 'Erz'< / span > };
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< / pre >
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< p >
The minus sine is put here because there is already a minus sign included due to the computation of the position error.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > load(< span class = "org-string" > 'mat/stages.mat'< / span > , < span class = "org-string" > 'nano_hexapod'< / span > );
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Gx = < span class = "org-type" > -< / span > G< span class = "org-type" > *< / span > inv(nano_hexapod.kinematics.J< span class = "org-type" > '< / span > );
Gx.InputName = {< span class = "org-string" > 'Fx'< / span > , < span class = "org-string" > 'Fy'< / span > , < span class = "org-string" > 'Fz'< / span > , < span class = "org-string" > 'Mx'< / span > , < span class = "org-string" > 'My'< / span > , < span class = "org-string" > 'Mz'< / span > };
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org2a269d5" class = "outline-3" >
< h3 id = "org2a269d5" > < span class = "section-number-3" > 4.2< / span > Controller Design< / h3 >
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< div class = "outline-text-3" id = "text-4-2" >
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< p >
The controller consists of:
< / p >
< ul class = "org-ul" >
< li > A pure integrator< / li >
< li > A Second integrator up to half the wanted bandwidth< / li >
< li > A Lead around the cross-over frequency< / li >
< li > A low pass filter with a cut-off equal to two times the wanted bandwidth< / li >
< / ul >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > wc = 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > 15; < span class = "org-comment" > % Bandwidth Bandwidth [rad/s]< / span >
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h = 1.5; < span class = "org-comment" > % Lead parameter< / span >
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Kx = (1< span class = "org-type" > /< / span > h) < span class = "org-type" > *< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > wc< span class = "org-type" > *< / span > h)< span class = "org-type" > /< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > wc< span class = "org-type" > /< / span > h) < span class = "org-type" > *< / span > wc< span class = "org-type" > /< / span > s < span class = "org-type" > *< / span > ((s< span class = "org-type" > /< / span > wc< span class = "org-type" > *< / span > 2 < span class = "org-type" > +< / span > 1)< span class = "org-type" > /< / span > (s< span class = "org-type" > /< / span > wc< span class = "org-type" > *< / span > 2)) < span class = "org-type" > *< / span > (1< span class = "org-type" > /< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > wc< span class = "org-type" > /< / span > 2));
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< span class = "org-comment" > % Normalization of the gain of have a loop gain of 1 at frequency wc< / span >
Kx = Kx< span class = "org-type" > .*< / span > diag(1< span class = "org-type" > ./< / span > diag(abs(freqresp(Gx< span class = "org-type" > *< / span > Kx, wc))));
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > isstable(feedback(Gx< span class = "org-type" > *< / span > Kx, eye(6), < span class = "org-type" > -< / span > 1))
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > Kx = inv(nano_hexapod.kinematics.J< span class = "org-type" > '< / span > )< span class = "org-type" > *< / span > Kx;
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > isstable(feedback(G< span class = "org-type" > *< / span > Kx, eye(6), 1))
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< / pre >
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< div id = "outline-container-org1171960" class = "outline-2" >
< h2 id = "org1171960" > < span class = "section-number-2" > 5< / span > Simulation< / h2 >
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< div class = "outline-text-2" id = "text-5" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > load(< span class = "org-string" > 'mat/conf_simulink.mat'< / span > );
< span class = "org-matlab-simulink-keyword" > set_param< / span > (< span class = "org-variable-name" > conf_simulink< / span > , < span class = "org-string" > 'StopTime'< / span > , < span class = "org-string" > '2'< / span > );
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< p >
And we simulate the system.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-matlab-simulink-keyword" > sim< / span > (< span class = "org-string" > 'nass_model'< / span > );
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > hac_dvf = simout;
save(< span class = "org-string" > './mat/tomo_exp_hac_lac.mat'< / span > , < span class = "org-string" > 'hac_dvf'< / span > );
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< div id = "outline-container-orgad3e1c7" class = "outline-2" >
< h2 id = "orgad3e1c7" > < span class = "section-number-2" > 6< / span > Results< / h2 >
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< div class = "outline-text-2" id = "text-6" >
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< p >
Let’ s load the simulation when no control is applied.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > load(< span class = "org-string" > './mat/experiment_tomography.mat'< / span > , < span class = "org-string" > 'tomo_align_dist'< / span > );
load(< span class = "org-string" > './mat/tomo_exp_hac_lac.mat'< / span > , < span class = "org-string" > 'hac_dvf'< / span > );
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< / pre >
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< / div >
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< div id = "postamble" class = "status" >
< p class = "author" > Author: Dehaeze Thomas< / p >
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< p class = "date" > Created: 2021-02-20 sam. 23:08< / p >
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< / body >
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