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< body >
< div id = "content" >
< h1 class = "title" > Cercalo Test Bench< / 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 = "#orgff87b8d" > 1. Introduction< / a > < / li >
< li > < a href = "#org2459267" > 2. Identification< / a >
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< ul >
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< li > < a href = "#org9b7ddaf" > 2.1. Excitation Data< / a > < / li >
< li > < a href = "#org04e353b" > 2.2. Huddle Test< / a > < / li >
< li > < a href = "#org4dc3ca0" > 2.3. Input / Output data< / a > < / li >
< li > < a href = "#org1f732b6" > 2.4. Estimation of the Frequency Response Function Matrix< / a > < / li >
< li > < a href = "#org8986a6a" > 2.5. Coherence< / a > < / li >
< li > < a href = "#orgb611053" > 2.6. Extraction of a transfer function matrix< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org5cfd6fc" > 3. Sensor Noise< / a > < / li >
< li > < a href = "#org2bd416c" > 4. Plant Analysis< / a >
< ul >
< li > < a href = "#orge74c87e" > 4.1. Rotation Matrix< / a > < / li >
< / ul >
< / li >
< li > < a href = "#org7b9c8d0" > 5. Control Objective< / a > < / li >
< li > < a href = "#orgbefa210" > 6. Plant Scaling< / a > < / li >
< li > < a href = "#org90e1042" > 7. Control Design< / a > < / li >
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< / ul >
< / div >
< / div >
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< div id = "outline-container-orgff87b8d" class = "outline-2" >
< h2 id = "orgff87b8d" > < span class = "section-number-2" > 1< / span > Introduction< / h2 >
< / div >
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< div id = "outline-container-org2459267" class = "outline-2" >
< h2 id = "org2459267" > < span class = "section-number-2" > 2< / span > Identification< / h2 >
< div class = "outline-text-2" id = "text-2" >
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< p >
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< a id = "org1f481c7" > < / a >
2019-09-10 18:18:28 +02:00
< / p >
< div class = "note" >
< p >
All the files (data and Matlab scripts) are accessible < a href = "data/plant_identification.zip" > here< / a > .
< / p >
< / div >
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< / div >
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< div id = "outline-container-org9b7ddaf" class = "outline-3" >
< h3 id = "org9b7ddaf" > < span class = "section-number-3" > 2.1< / span > Excitation Data< / h3 >
< div class = "outline-text-3" id = "text-2-1" >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > fs = < span class = "org-highlight-numbers-number" > 1e4< / span > ;
Ts = < span class = "org-highlight-numbers-number" > 1< / span > < span class = "org-type" > /< / span > fs;
< / pre >
< / div >
< p >
We generate white noise with the "random number" simulink block, and we filter that noise.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Gi = < span class = "org-rainbow-delimiters-depth-1" > (< / span > < span class = "org-highlight-numbers-number" > 1< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > < span class = "org-type" > /< / span > < span class = "org-rainbow-delimiters-depth-1" > (< / span > < span class = "org-highlight-numbers-number" > 1< / span > < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > < span class = "org-highlight-numbers-number" > 2< / span > < span class = "org-type" > /< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > /< / span > < span class = "org-highlight-numbers-number" > 100< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > c2d< span class = "org-rainbow-delimiters-depth-1" > (< / span > Gi, Ts, < span class = "org-string" > 'tustin'< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span >
< / pre >
< / div >
< pre class = "example" >
c2d(Gi, Ts, 'tustin')
ans =
0.030459 (z+1)
--------------
(z-0.9391)
Sample time: 0.0001 seconds
Discrete-time zero/pole/gain model.
< / pre >
< / div >
< / div >
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< div id = "outline-container-org04e353b" class = "outline-3" >
< h3 id = "org04e353b" > < span class = "section-number-3" > 2.2< / span > Huddle Test< / h3 >
< div class = "outline-text-3" id = "text-2-2" >
< p >
We load the data taken during the Huddle Test.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > load< span class = "org-rainbow-delimiters-depth-1" > (< / span > < span class = "org-string" > 'mat/data_huddle_test.mat', 't', 'xh', 'xv', 'cuh', 'cuv'< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
< / pre >
< / div >
< p >
The variables are:
< / p >
< dl class = "org-dl" >
< dt > \(x_h\)< / dt > < dd > Normalized position of the beam in the horizontal direction as measured by the 4 quadrant diode< / dd >
< dt > \(x_v\)< / dt > < dd > Normalized position of the beam in the vertical direction as measured by the 4 quadrant diode< / dd >
< dt > \(cu_h\)< / dt > < dd > Voltage across the inductance used for the horizontal positioning of the Cercalo< / dd >
< dt > \(vu_v\)< / dt > < dd > Voltage across the inductance used for the vertical positioning of the Cercalo< / dd >
< / dl >
< div class = "org-src-container" >
< pre class = "src src-matlab" > xh< span class = "org-rainbow-delimiters-depth-1" > (< / span > t< span class = "org-type" > < < / span > < span class = "org-highlight-numbers-number" > 1< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > = < span class = "org-rainbow-delimiters-depth-1" > []< / span > ;
xv< span class = "org-rainbow-delimiters-depth-1" > (< / span > t< span class = "org-type" > < < / span > < span class = "org-highlight-numbers-number" > 1< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > = < span class = "org-rainbow-delimiters-depth-1" > []< / span > ;
cuh< span class = "org-rainbow-delimiters-depth-1" > (< / span > t< span class = "org-type" > < < / span > < span class = "org-highlight-numbers-number" > 1< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > = < span class = "org-rainbow-delimiters-depth-1" > []< / span > ;
cuv< span class = "org-rainbow-delimiters-depth-1" > (< / span > t< span class = "org-type" > < < / span > < span class = "org-highlight-numbers-number" > 1< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > = < span class = "org-rainbow-delimiters-depth-1" > []< / span > ;
t< span class = "org-rainbow-delimiters-depth-1" > (< / span > t< span class = "org-type" > < < / span > < span class = "org-highlight-numbers-number" > 1< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > = < span class = "org-rainbow-delimiters-depth-1" > []< / span > ;
t = t < span class = "org-type" > -< / span > t< span class = "org-rainbow-delimiters-depth-1" > (< / span > < span class = "org-highlight-numbers-number" > 1< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-rainbow-delimiters-depth-1" > [< / span > psd_xh, f< span class = "org-rainbow-delimiters-depth-1" > ]< / span > = pwelch< span class = "org-rainbow-delimiters-depth-1" > (< / span > xh, hanning< span class = "org-rainbow-delimiters-depth-2" > (< / span > ceil< span class = "org-rainbow-delimiters-depth-3" > (< / span > < span class = "org-highlight-numbers-number" > 1< / span > < span class = "org-type" > *< / span > fs< span class = "org-rainbow-delimiters-depth-3" > )< / span > < span class = "org-rainbow-delimiters-depth-2" > )< / span > , < span class = "org-rainbow-delimiters-depth-2" > []< / span > , < span class = "org-rainbow-delimiters-depth-2" > []< / span > , fs< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
< span class = "org-rainbow-delimiters-depth-1" > [< / span > psd_xv, < span class = "org-type" > ~< / span > < span class = "org-rainbow-delimiters-depth-1" > ]< / span > = pwelch< span class = "org-rainbow-delimiters-depth-1" > (< / span > xv, hanning< span class = "org-rainbow-delimiters-depth-2" > (< / span > ceil< span class = "org-rainbow-delimiters-depth-3" > (< / span > < span class = "org-highlight-numbers-number" > 1< / span > < span class = "org-type" > *< / span > fs< span class = "org-rainbow-delimiters-depth-3" > )< / span > < span class = "org-rainbow-delimiters-depth-2" > )< / span > , < span class = "org-rainbow-delimiters-depth-2" > []< / span > , < span class = "org-rainbow-delimiters-depth-2" > []< / span > , fs< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-type" > figure< / span > ;
hold on;
plot< span class = "org-rainbow-delimiters-depth-1" > (< / span > f, sqrt< span class = "org-rainbow-delimiters-depth-2" > (< / span > psd_xh< span class = "org-rainbow-delimiters-depth-2" > )< / span > , < span class = "org-string" > 'DisplayName', '< / span > $< span class = "org-type" > \< / span > Gamma_< span class = "org-rainbow-delimiters-depth-2" > {< / span > x_h< span class = "org-rainbow-delimiters-depth-2" > }< / span > $'< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
plot< span class = "org-rainbow-delimiters-depth-1" > (< / span > f, sqrt< span class = "org-rainbow-delimiters-depth-2" > (< / span > psd_xv< span class = "org-rainbow-delimiters-depth-2" > )< / span > , < span class = "org-string" > 'DisplayName', '< / span > $< span class = "org-type" > \< / span > Gamma_< span class = "org-rainbow-delimiters-depth-2" > {< / span > x_v< span class = "org-rainbow-delimiters-depth-2" > }< / span > $'< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
hold off;
< span class = "org-type" > set< / span > < span class = "org-rainbow-delimiters-depth-1" > (< / span > < span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'xscale', 'log'< / span > < span class = "org-string" > < span class = "org-rainbow-delimiters-depth-1" > )< / span > < / span > < span class = "org-string" > ; set< / span > < span class = "org-string" > < span class = "org-rainbow-delimiters-depth-1" > (< / span > < / span > < span class = "org-string" > gca, 'yscale', 'log'< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
xlabel< span class = "org-rainbow-delimiters-depth-1" > (< / span > < span class = "org-string" > 'Frequency < / span > < span class = "org-string" > < span class = "org-rainbow-delimiters-depth-2" > [< / span > < / span > < span class = "org-string" > Hz< / span > < span class = "org-string" > < span class = "org-rainbow-delimiters-depth-2" > ]< / span > < / span > < span class = "org-string" > '< / span > < span class = "org-string" > < span class = "org-rainbow-delimiters-depth-1" > )< / span > < / span > < span class = "org-string" > ; ylabel< / span > < span class = "org-string" > < span class = "org-rainbow-delimiters-depth-1" > (< / span > < / span > < span class = "org-string" > '< / span > ASD $< span class = "org-type" > \< / span > left< span class = "org-rainbow-delimiters-depth-2" > [< / span > < span class = "org-type" > \< / span > frac< span class = "org-rainbow-delimiters-depth-3" > {< / span > V< span class = "org-rainbow-delimiters-depth-3" > }{< / span > < span class = "org-type" > \< / span > sqrt< span class = "org-rainbow-delimiters-depth-4" > {< / span > Hz< span class = "org-rainbow-delimiters-depth-4" > }< / span > < span class = "org-rainbow-delimiters-depth-3" > }< / span > < span class = "org-type" > \< / span > right< span class = "org-rainbow-delimiters-depth-2" > ]< / span > $'< span class = "org-rainbow-delimiters-depth-1" > )< / span >
legend< span class = "org-rainbow-delimiters-depth-1" > (< / span > < span class = "org-string" > 'Location', 'southwest'< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
xlim< span class = "org-rainbow-delimiters-depth-1" > (< / span > < span class = "org-rainbow-delimiters-depth-2" > [< / span > < span class = "org-highlight-numbers-number" > 1< / span > , < span class = "org-highlight-numbers-number" > 1000< / span > < span class = "org-rainbow-delimiters-depth-2" > ]< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-rainbow-delimiters-depth-1" > [< / span > psd_cuh, f< span class = "org-rainbow-delimiters-depth-1" > ]< / span > = pwelch< span class = "org-rainbow-delimiters-depth-1" > (< / span > cuh, hanning< span class = "org-rainbow-delimiters-depth-2" > (< / span > ceil< span class = "org-rainbow-delimiters-depth-3" > (< / span > < span class = "org-highlight-numbers-number" > 1< / span > < span class = "org-type" > *< / span > fs< span class = "org-rainbow-delimiters-depth-3" > )< / span > < span class = "org-rainbow-delimiters-depth-2" > )< / span > , < span class = "org-rainbow-delimiters-depth-2" > []< / span > , < span class = "org-rainbow-delimiters-depth-2" > []< / span > , fs< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
< span class = "org-rainbow-delimiters-depth-1" > [< / span > psd_cuv, < span class = "org-type" > ~< / span > < span class = "org-rainbow-delimiters-depth-1" > ]< / span > = pwelch< span class = "org-rainbow-delimiters-depth-1" > (< / span > cuv, hanning< span class = "org-rainbow-delimiters-depth-2" > (< / span > ceil< span class = "org-rainbow-delimiters-depth-3" > (< / span > < span class = "org-highlight-numbers-number" > 1< / span > < span class = "org-type" > *< / span > fs< span class = "org-rainbow-delimiters-depth-3" > )< / span > < span class = "org-rainbow-delimiters-depth-2" > )< / span > , < span class = "org-rainbow-delimiters-depth-2" > []< / span > , < span class = "org-rainbow-delimiters-depth-2" > []< / span > , fs< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-type" > figure< / span > ;
hold on;
plot< span class = "org-rainbow-delimiters-depth-1" > (< / span > f, sqrt< span class = "org-rainbow-delimiters-depth-2" > (< / span > psd_cuh< span class = "org-rainbow-delimiters-depth-2" > )< / span > , < span class = "org-string" > 'DisplayName', '< / span > $< span class = "org-type" > \< / span > Gamma_< span class = "org-rainbow-delimiters-depth-2" > {< / span > cu_h< span class = "org-rainbow-delimiters-depth-2" > }< / span > $'< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
plot< span class = "org-rainbow-delimiters-depth-1" > (< / span > f, sqrt< span class = "org-rainbow-delimiters-depth-2" > (< / span > psd_cuv< span class = "org-rainbow-delimiters-depth-2" > )< / span > , < span class = "org-string" > 'DisplayName', '< / span > $< span class = "org-type" > \< / span > Gamma_< span class = "org-rainbow-delimiters-depth-2" > {< / span > cu_v< span class = "org-rainbow-delimiters-depth-2" > }< / span > $'< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
hold off;
< span class = "org-type" > set< / span > < span class = "org-rainbow-delimiters-depth-1" > (< / span > < span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'xscale', 'log'< / span > < span class = "org-string" > < span class = "org-rainbow-delimiters-depth-1" > )< / span > < / span > < span class = "org-string" > ; set< / span > < span class = "org-string" > < span class = "org-rainbow-delimiters-depth-1" > (< / span > < / span > < span class = "org-string" > gca, 'yscale', 'log'< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
xlabel< span class = "org-rainbow-delimiters-depth-1" > (< / span > < span class = "org-string" > 'Frequency < / span > < span class = "org-string" > < span class = "org-rainbow-delimiters-depth-2" > [< / span > < / span > < span class = "org-string" > Hz< / span > < span class = "org-string" > < span class = "org-rainbow-delimiters-depth-2" > ]< / span > < / span > < span class = "org-string" > '< / span > < span class = "org-string" > < span class = "org-rainbow-delimiters-depth-1" > )< / span > < / span > < span class = "org-string" > ; ylabel< / span > < span class = "org-string" > < span class = "org-rainbow-delimiters-depth-1" > (< / span > < / span > < span class = "org-string" > '< / span > ASD $< span class = "org-type" > \< / span > left< span class = "org-rainbow-delimiters-depth-2" > [< / span > < span class = "org-type" > \< / span > frac< span class = "org-rainbow-delimiters-depth-3" > {< / span > V< span class = "org-rainbow-delimiters-depth-3" > }{< / span > < span class = "org-type" > \< / span > sqrt< span class = "org-rainbow-delimiters-depth-4" > {< / span > Hz< span class = "org-rainbow-delimiters-depth-4" > }< / span > < span class = "org-rainbow-delimiters-depth-3" > }< / span > < span class = "org-type" > \< / span > right< span class = "org-rainbow-delimiters-depth-2" > ]< / span > $'< span class = "org-rainbow-delimiters-depth-1" > )< / span >
legend< span class = "org-rainbow-delimiters-depth-1" > (< / span > < span class = "org-string" > 'Location', 'southwest'< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
xlim< span class = "org-rainbow-delimiters-depth-1" > (< / span > < span class = "org-rainbow-delimiters-depth-2" > [< / span > < span class = "org-highlight-numbers-number" > 1< / span > , < span class = "org-highlight-numbers-number" > 1000< / span > < span class = "org-rainbow-delimiters-depth-2" > ]< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-org4dc3ca0" class = "outline-3" >
< h3 id = "org4dc3ca0" > < span class = "section-number-3" > 2.3< / span > Input / Output data< / h3 >
< div class = "outline-text-3" id = "text-2-3" >
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< p >
The identification data is loaded
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > ux = load< span class = "org-rainbow-delimiters-depth-1" > (< / span > < span class = "org-string" > 'mat/data_ux.mat', 't', 'ux', 'yx', 'yy'< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
uy = load< span class = "org-rainbow-delimiters-depth-1" > (< / span > < span class = "org-string" > 'mat/data_uy.mat', 't', 'uy', 'yx', 'yy'< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
2019-09-10 18:16:16 +02:00
< / pre >
< / div >
< p >
We remove the first seconds where the Cercalo is turned on.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > i0x = < span class = "org-highlight-numbers-number" > 20< / span > < span class = "org-type" > *< / span > fs;
i0y = < span class = "org-highlight-numbers-number" > 10< / span > < span class = "org-type" > *< / span > fs;
ux.t = ux.t< span class = "org-rainbow-delimiters-depth-1" > (< / span > i0x< span class = "org-type" > :< / span > end< span class = "org-rainbow-delimiters-depth-1" > )< / span > < span class = "org-type" > -< / span > ux.t< span class = "org-rainbow-delimiters-depth-1" > (< / span > i0x< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
ux.ux = ux.ux< span class = "org-rainbow-delimiters-depth-1" > (< / span > i0x< span class = "org-type" > :< / span > end< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
ux.yx = ux.yx< span class = "org-rainbow-delimiters-depth-1" > (< / span > i0x< span class = "org-type" > :< / span > end< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
ux.yy = ux.yy< span class = "org-rainbow-delimiters-depth-1" > (< / span > i0x< span class = "org-type" > :< / span > end< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
uy.t = uy.t< span class = "org-rainbow-delimiters-depth-1" > (< / span > i0y< span class = "org-type" > :< / span > end< span class = "org-rainbow-delimiters-depth-1" > )< / span > < span class = "org-type" > -< / span > uy.t< span class = "org-rainbow-delimiters-depth-1" > (< / span > i0x< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
uy.uy = uy.uy< span class = "org-rainbow-delimiters-depth-1" > (< / span > i0y< span class = "org-type" > :< / span > end< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
uy.yx = uy.yx< span class = "org-rainbow-delimiters-depth-1" > (< / span > i0y< span class = "org-type" > :< / span > end< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
uy.yy = uy.yy< span class = "org-rainbow-delimiters-depth-1" > (< / span > i0y< span class = "org-type" > :< / span > end< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > ux.ux = ux.ux< span class = "org-type" > -< / span > mean< span class = "org-rainbow-delimiters-depth-1" > (< / span > ux.ux< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
ux.yx = ux.yx< span class = "org-type" > -< / span > mean< span class = "org-rainbow-delimiters-depth-1" > (< / span > ux.yx< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
ux.yy = ux.yy< span class = "org-type" > -< / span > mean< span class = "org-rainbow-delimiters-depth-1" > (< / span > ux.yy< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
uy.ux = uy.ux< span class = "org-type" > -< / span > mean< span class = "org-rainbow-delimiters-depth-1" > (< / span > uy.ux< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
uy.yx = uy.yx< span class = "org-type" > -< / span > mean< span class = "org-rainbow-delimiters-depth-1" > (< / span > uy.yx< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
uy.yy = uy.yy< span class = "org-type" > -< / span > mean< span class = "org-rainbow-delimiters-depth-1" > (< / span > uy.yy< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
< / pre >
< / div >
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< div id = "org2d6aeb1" class = "figure" >
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< p > < img src = "figs/identification_ux.png" alt = "identification_ux.png" / >
< / p >
< p > < span class = "figure-number" > Figure 1: < / span > Identification signals when exciting the \(x\) axis (< a href = "./figs/identification_ux.png" > png< / a > , < a href = "./figs/identification_ux.pdf" > pdf< / a > )< / p >
< / div >
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< div id = "orgd5e71a4" class = "figure" >
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< p > < img src = "figs/identification_uy.png" alt = "identification_uy.png" / >
< / p >
< p > < span class = "figure-number" > Figure 2: < / span > Identification signals when exciting the \(y\) axis (< a href = "./figs/identification_uy.png" > png< / a > , < a href = "./figs/identification_uy.pdf" > pdf< / a > )< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-org1f732b6" class = "outline-3" >
< h3 id = "org1f732b6" > < span class = "section-number-3" > 2.4< / span > Estimation of the Frequency Response Function Matrix< / h3 >
< div class = "outline-text-3" id = "text-2-4" >
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< p >
We compute an estimate of the transfer functions.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-rainbow-delimiters-depth-1" > [< / span > tf_ux_yx, f< span class = "org-rainbow-delimiters-depth-1" > ]< / span > = tfestimate< span class = "org-rainbow-delimiters-depth-1" > (< / span > ux.ux, ux.yx, hanning< span class = "org-rainbow-delimiters-depth-2" > (< / span > ceil< span class = "org-rainbow-delimiters-depth-3" > (< / span > < span class = "org-highlight-numbers-number" > 1< / span > < span class = "org-type" > *< / span > fs< span class = "org-rainbow-delimiters-depth-3" > )< / span > < span class = "org-rainbow-delimiters-depth-2" > )< / span > , < span class = "org-rainbow-delimiters-depth-2" > []< / span > , < span class = "org-rainbow-delimiters-depth-2" > []< / span > , fs< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
< span class = "org-rainbow-delimiters-depth-1" > [< / span > tf_ux_yy, < span class = "org-type" > ~< / span > < span class = "org-rainbow-delimiters-depth-1" > ]< / span > = tfestimate< span class = "org-rainbow-delimiters-depth-1" > (< / span > ux.ux, ux.yy, hanning< span class = "org-rainbow-delimiters-depth-2" > (< / span > ceil< span class = "org-rainbow-delimiters-depth-3" > (< / span > < span class = "org-highlight-numbers-number" > 1< / span > < span class = "org-type" > *< / span > fs< span class = "org-rainbow-delimiters-depth-3" > )< / span > < span class = "org-rainbow-delimiters-depth-2" > )< / span > , < span class = "org-rainbow-delimiters-depth-2" > []< / span > , < span class = "org-rainbow-delimiters-depth-2" > []< / span > , fs< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
< span class = "org-rainbow-delimiters-depth-1" > [< / span > tf_uy_yx, < span class = "org-type" > ~< / span > < span class = "org-rainbow-delimiters-depth-1" > ]< / span > = tfestimate< span class = "org-rainbow-delimiters-depth-1" > (< / span > uy.uy, uy.yx, hanning< span class = "org-rainbow-delimiters-depth-2" > (< / span > ceil< span class = "org-rainbow-delimiters-depth-3" > (< / span > < span class = "org-highlight-numbers-number" > 1< / span > < span class = "org-type" > *< / span > fs< span class = "org-rainbow-delimiters-depth-3" > )< / span > < span class = "org-rainbow-delimiters-depth-2" > )< / span > , < span class = "org-rainbow-delimiters-depth-2" > []< / span > , < span class = "org-rainbow-delimiters-depth-2" > []< / span > , fs< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
< span class = "org-rainbow-delimiters-depth-1" > [< / span > tf_uy_yy, < span class = "org-type" > ~< / span > < span class = "org-rainbow-delimiters-depth-1" > ]< / span > = tfestimate< span class = "org-rainbow-delimiters-depth-1" > (< / span > uy.uy, uy.yy, hanning< span class = "org-rainbow-delimiters-depth-2" > (< / span > ceil< span class = "org-rainbow-delimiters-depth-3" > (< / span > < span class = "org-highlight-numbers-number" > 1< / span > < span class = "org-type" > *< / span > fs< span class = "org-rainbow-delimiters-depth-3" > )< / span > < span class = "org-rainbow-delimiters-depth-2" > )< / span > , < span class = "org-rainbow-delimiters-depth-2" > []< / span > , < span class = "org-rainbow-delimiters-depth-2" > []< / span > , fs< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
< / pre >
< / div >
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< div id = "org2051377" class = "figure" >
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< p > < img src = "figs/frequency_response_matrix.png" alt = "frequency_response_matrix.png" / >
< / p >
< p > < span class = "figure-number" > Figure 3: < / span > Frequency Response Matrix (< a href = "./figs/frequency_response_matrix.png" > png< / a > , < a href = "./figs/frequency_response_matrix.pdf" > pdf< / a > )< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-org8986a6a" class = "outline-3" >
< h3 id = "org8986a6a" > < span class = "section-number-3" > 2.5< / span > Coherence< / h3 >
< div class = "outline-text-3" id = "text-2-5" >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-rainbow-delimiters-depth-1" > [< / span > coh_ux_yx, f< span class = "org-rainbow-delimiters-depth-1" > ]< / span > = mscohere< span class = "org-rainbow-delimiters-depth-1" > (< / span > ux.ux, ux.yx, hanning< span class = "org-rainbow-delimiters-depth-2" > (< / span > ceil< span class = "org-rainbow-delimiters-depth-3" > (< / span > < span class = "org-highlight-numbers-number" > 1< / span > < span class = "org-type" > *< / span > fs< span class = "org-rainbow-delimiters-depth-3" > )< / span > < span class = "org-rainbow-delimiters-depth-2" > )< / span > , < span class = "org-rainbow-delimiters-depth-2" > []< / span > , < span class = "org-rainbow-delimiters-depth-2" > []< / span > , fs< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
< span class = "org-rainbow-delimiters-depth-1" > [< / span > coh_ux_yy, < span class = "org-type" > ~< / span > < span class = "org-rainbow-delimiters-depth-1" > ]< / span > = mscohere< span class = "org-rainbow-delimiters-depth-1" > (< / span > ux.ux, ux.yy, hanning< span class = "org-rainbow-delimiters-depth-2" > (< / span > ceil< span class = "org-rainbow-delimiters-depth-3" > (< / span > < span class = "org-highlight-numbers-number" > 1< / span > < span class = "org-type" > *< / span > fs< span class = "org-rainbow-delimiters-depth-3" > )< / span > < span class = "org-rainbow-delimiters-depth-2" > )< / span > , < span class = "org-rainbow-delimiters-depth-2" > []< / span > , < span class = "org-rainbow-delimiters-depth-2" > []< / span > , fs< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
< span class = "org-rainbow-delimiters-depth-1" > [< / span > coh_uy_yx, < span class = "org-type" > ~< / span > < span class = "org-rainbow-delimiters-depth-1" > ]< / span > = mscohere< span class = "org-rainbow-delimiters-depth-1" > (< / span > uy.uy, uy.yx, hanning< span class = "org-rainbow-delimiters-depth-2" > (< / span > ceil< span class = "org-rainbow-delimiters-depth-3" > (< / span > < span class = "org-highlight-numbers-number" > 1< / span > < span class = "org-type" > *< / span > fs< span class = "org-rainbow-delimiters-depth-3" > )< / span > < span class = "org-rainbow-delimiters-depth-2" > )< / span > , < span class = "org-rainbow-delimiters-depth-2" > []< / span > , < span class = "org-rainbow-delimiters-depth-2" > []< / span > , fs< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
< span class = "org-rainbow-delimiters-depth-1" > [< / span > coh_uy_yy, < span class = "org-type" > ~< / span > < span class = "org-rainbow-delimiters-depth-1" > ]< / span > = mscohere< span class = "org-rainbow-delimiters-depth-1" > (< / span > uy.uy, uy.yy, hanning< span class = "org-rainbow-delimiters-depth-2" > (< / span > ceil< span class = "org-rainbow-delimiters-depth-3" > (< / span > < span class = "org-highlight-numbers-number" > 1< / span > < span class = "org-type" > *< / span > fs< span class = "org-rainbow-delimiters-depth-3" > )< / span > < span class = "org-rainbow-delimiters-depth-2" > )< / span > , < span class = "org-rainbow-delimiters-depth-2" > []< / span > , < span class = "org-rainbow-delimiters-depth-2" > []< / span > , fs< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
< / pre >
< / div >
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< div id = "orgf0f9ad1" class = "figure" >
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< p > < img src = "figs/identification_coherence.png" alt = "identification_coherence.png" / >
< / p >
< p > < span class = "figure-number" > Figure 4: < / span > Coherence (< a href = "./figs/identification_coherence.png" > png< / a > , < a href = "./figs/identification_coherence.pdf" > pdf< / a > )< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgb611053" class = "outline-3" >
< h3 id = "orgb611053" > < span class = "section-number-3" > 2.6< / span > Extraction of a transfer function matrix< / h3 >
< div class = "outline-text-3" id = "text-2-6" >
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< p >
First we define the initial guess for the resonance frequencies and the weights associated.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > freqs_res = < span class = "org-rainbow-delimiters-depth-1" > [< / span > < span class = "org-highlight-numbers-number" > 410< / span > , < span class = "org-highlight-numbers-number" > 250< / span > < span class = "org-rainbow-delimiters-depth-1" > ]< / span > ; < span class = "org-comment" > % [Hz]< / span >
freqs_res_weights = < span class = "org-rainbow-delimiters-depth-1" > [< / span > < span class = "org-highlight-numbers-number" > 10< / span > , < span class = "org-highlight-numbers-number" > 10< / span > < span class = "org-rainbow-delimiters-depth-1" > ]< / span > ; < span class = "org-comment" > % [Hz]< / span >
< / pre >
< / div >
< p >
From the number of resonance frequency we want to fit, we define the order < code > N< / code > of the system we want to obtain.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > N = < span class = "org-highlight-numbers-number" > 2< / span > < span class = "org-type" > *< / span > length< span class = "org-rainbow-delimiters-depth-1" > (< / span > freqs_res< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
< / pre >
< / div >
< p >
We then make an initial guess on the complex values of the poles.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > xi = < span class = "org-highlight-numbers-number" > 0< / span > .< span class = "org-highlight-numbers-number" > 001< / span > ; < span class = "org-comment" > % Approximate modal damping< / span >
poles = < span class = "org-rainbow-delimiters-depth-1" > [< / span > < span class = "org-highlight-numbers-number" > 2< / span > < span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > freqs_res< span class = "org-type" > *< / span > < span class = "org-rainbow-delimiters-depth-2" > (< / span > xi < span class = "org-type" > +< / span > < span class = "org-highlight-numbers-number" > 1i< / span > < span class = "org-rainbow-delimiters-depth-2" > )< / span > , < span class = "org-highlight-numbers-number" > 2< / span > < span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > freqs_res< span class = "org-type" > *< / span > < span class = "org-rainbow-delimiters-depth-2" > (< / span > xi < span class = "org-type" > -< / span > < span class = "org-highlight-numbers-number" > 1i< / span > < span class = "org-rainbow-delimiters-depth-2" > )< / span > < span class = "org-rainbow-delimiters-depth-1" > ]< / span > ;
< / pre >
< / div >
< p >
We then define the weight that will be used for the fitting.
Basically, we want more weight around the resonance and at low frequency (below the first resonance).
Also, we want more importance where we have a better coherence.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > weight = ones< span class = "org-rainbow-delimiters-depth-1" > (< / span > < span class = "org-highlight-numbers-number" > 1< / span > , length< span class = "org-rainbow-delimiters-depth-2" > (< / span > f< span class = "org-rainbow-delimiters-depth-2" > )< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
< span class = "org-comment" > % weight = G_coh';< / span >
% alpha = < span class = "org-highlight-numbers-number" > 0< / span > .< span class = "org-highlight-numbers-number" > 1< / span > ;
% < span class = "org-keyword" > for< / span > < span class = "org-variable-name" > freq_i< / span > = < span class = "org-constant" > < span class = "org-highlight-numbers-number" > 1< / span > < / span > < span class = "org-constant" > :length(freqs_res)< / span >
% weight(f< span class = "org-type" > > < / span > (< span class = "org-highlight-numbers-number" > 1< / span > < span class = "org-type" > -< / span > alpha)< span class = "org-type" > *< / span > freqs_res(freq_i) < span class = "org-type" > & < / span > omega< span class = "org-type" > < < / span > (< span class = "org-highlight-numbers-number" > 1< / span > < span class = "org-type" > +< / span > alpha)< span class = "org-type" > *< / span > < span class = "org-highlight-numbers-number" > 2< / span > < span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > freqs_res(freq_i)) = freqs_res_weights(freq_i);
< span class = "org-comment" > % end< / span >
< / pre >
< / div >
< p >
Ignore data above some frequency.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > weight< span class = "org-rainbow-delimiters-depth-1" > (< / span > f< span class = "org-type" > > < / span > < span class = "org-highlight-numbers-number" > 1000< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > = < span class = "org-highlight-numbers-number" > 0< / span > ;
< / pre >
< / div >
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< div id = "org9d870b3" class = "figure" >
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< p > < img src = "figs/weights.png" alt = "weights.png" / >
< / p >
< p > < span class = "figure-number" > Figure 5: < / span > Weights amplitude (< a href = "./figs/weights.png" > png< / a > , < a href = "./figs/weights.pdf" > pdf< / a > )< / p >
< / div >
< p >
When we set some options for < code > vfit3< / code > .
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > opts = struct< span class = "org-rainbow-delimiters-depth-1" > ()< / span > ;
opts.stable = < span class = "org-highlight-numbers-number" > 1< / span > ; < span class = "org-comment" > % Enforce stable poles< / span >
opts.asymp = < span class = "org-highlight-numbers-number" > 1< / span > ; < span class = "org-comment" > % Force D matrix to be null< / span >
opts.relax = < span class = "org-highlight-numbers-number" > 1< / span > ; < span class = "org-comment" > % Use vector fitting with relaxed non-triviality constraint< / span >
opts.skip_pole = < span class = "org-highlight-numbers-number" > 0< / span > ; < span class = "org-comment" > % Do NOT skip pole identification< / span >
opts.skip_res = < span class = "org-highlight-numbers-number" > 0< / span > ; < span class = "org-comment" > % Do NOT skip identification of residues (C,D,E)< / span >
opts.cmplx_ss = < span class = "org-highlight-numbers-number" > 0< / span > ; < span class = "org-comment" > % Create real state space model with block diagonal A< / span >
opts.spy1 = < span class = "org-highlight-numbers-number" > 0< / span > ; < span class = "org-comment" > % No plotting for first stage of vector fitting< / span >
opts.spy2 = < span class = "org-highlight-numbers-number" > 0< / span > ; < span class = "org-comment" > % Create magnitude plot for fitting of f(s)< / span >
< / pre >
< / div >
< p >
We define the number of iteration.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Niter = < span class = "org-highlight-numbers-number" > 5< / span > ;
< / pre >
< / div >
< p >
An we run the < code > vectfit3< / code > algorithm.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-keyword" > for< / span > < span class = "org-variable-name" > iter< / span > = < span class = "org-constant" > < span class = "org-highlight-numbers-number" > 1< / span > < / span > < span class = "org-constant" > :Niter< / span >
< span class = "org-rainbow-delimiters-depth-1" > [< / span > SER_ux_yx, poles, < span class = "org-type" > ~< / span > , fit_ux_yx< span class = "org-rainbow-delimiters-depth-1" > ]< / span > = vectfit3< span class = "org-rainbow-delimiters-depth-1" > (< / span > tf_ux_yx< span class = "org-type" > .'< / span > , < span class = "org-highlight-numbers-number" > 1i< / span > < span class = "org-type" > *< / span > < span class = "org-highlight-numbers-number" > 2< / span > < span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > f, poles, weight, opts< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
< span class = "org-keyword" > end< / span >
< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > iter< / span > = < span class = "org-constant" > < span class = "org-highlight-numbers-number" > 1< / span > < / span > < span class = "org-constant" > :Niter< / span >
< span class = "org-rainbow-delimiters-depth-1" > [< / span > SER_uy_yx, poles, < span class = "org-type" > ~< / span > , fit_uy_yx< span class = "org-rainbow-delimiters-depth-1" > ]< / span > = vectfit3< span class = "org-rainbow-delimiters-depth-1" > (< / span > tf_uy_yx< span class = "org-type" > .'< / span > , < span class = "org-highlight-numbers-number" > 1i< / span > < span class = "org-type" > *< / span > < span class = "org-highlight-numbers-number" > 2< / span > < span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > f, poles, weight, opts< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
< span class = "org-keyword" > end< / span >
< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > iter< / span > = < span class = "org-constant" > < span class = "org-highlight-numbers-number" > 1< / span > < / span > < span class = "org-constant" > :Niter< / span >
< span class = "org-rainbow-delimiters-depth-1" > [< / span > SER_ux_yy, poles, < span class = "org-type" > ~< / span > , fit_ux_yy< span class = "org-rainbow-delimiters-depth-1" > ]< / span > = vectfit3< span class = "org-rainbow-delimiters-depth-1" > (< / span > tf_ux_yy< span class = "org-type" > .'< / span > , < span class = "org-highlight-numbers-number" > 1i< / span > < span class = "org-type" > *< / span > < span class = "org-highlight-numbers-number" > 2< / span > < span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > f, poles, weight, opts< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
< span class = "org-keyword" > end< / span >
< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > iter< / span > = < span class = "org-constant" > < span class = "org-highlight-numbers-number" > 1< / span > < / span > < span class = "org-constant" > :Niter< / span >
< span class = "org-rainbow-delimiters-depth-1" > [< / span > SER_uy_yy, poles, < span class = "org-type" > ~< / span > , fit_uy_yy< span class = "org-rainbow-delimiters-depth-1" > ]< / span > = vectfit3< span class = "org-rainbow-delimiters-depth-1" > (< / span > tf_uy_yy< span class = "org-type" > .'< / span > , < span class = "org-highlight-numbers-number" > 1i< / span > < span class = "org-type" > *< / span > < span class = "org-highlight-numbers-number" > 2< / span > < span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > f, poles, weight, opts< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
< span class = "org-keyword" > end< / span >
< / pre >
< / div >
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< div id = "orgce7fc34" class = "figure" >
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< p > < img src = "figs/identification_matrix_fit.png" alt = "identification_matrix_fit.png" / >
< / p >
< p > < span class = "figure-number" > Figure 6: < / span > Transfer Function Extraction of the FRF matrix (< a href = "./figs/identification_matrix_fit.png" > png< / a > , < a href = "./figs/identification_matrix_fit.pdf" > pdf< / a > )< / p >
< / div >
< p >
And finally, we create the identified state space model:
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > G_ux_yx = minreal< span class = "org-rainbow-delimiters-depth-1" > (< / span > ss< span class = "org-rainbow-delimiters-depth-2" > (< / span > full< span class = "org-rainbow-delimiters-depth-3" > (< / span > SER_ux_yx.A< span class = "org-rainbow-delimiters-depth-3" > )< / span > ,SER_ux_yx.B,SER_ux_yx.C,SER_ux_yx.D< span class = "org-rainbow-delimiters-depth-2" > )< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
G_uy_yx = minreal< span class = "org-rainbow-delimiters-depth-1" > (< / span > ss< span class = "org-rainbow-delimiters-depth-2" > (< / span > full< span class = "org-rainbow-delimiters-depth-3" > (< / span > SER_uy_yx.A< span class = "org-rainbow-delimiters-depth-3" > )< / span > ,SER_uy_yx.B,SER_uy_yx.C,SER_uy_yx.D< span class = "org-rainbow-delimiters-depth-2" > )< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
G_ux_yy = minreal< span class = "org-rainbow-delimiters-depth-1" > (< / span > ss< span class = "org-rainbow-delimiters-depth-2" > (< / span > full< span class = "org-rainbow-delimiters-depth-3" > (< / span > SER_ux_yy.A< span class = "org-rainbow-delimiters-depth-3" > )< / span > ,SER_ux_yy.B,SER_ux_yy.C,SER_ux_yy.D< span class = "org-rainbow-delimiters-depth-2" > )< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
G_uy_yy = minreal< span class = "org-rainbow-delimiters-depth-1" > (< / span > ss< span class = "org-rainbow-delimiters-depth-2" > (< / span > full< span class = "org-rainbow-delimiters-depth-3" > (< / span > SER_uy_yy.A< span class = "org-rainbow-delimiters-depth-3" > )< / span > ,SER_uy_yy.B,SER_uy_yy.C,SER_uy_yy.D< span class = "org-rainbow-delimiters-depth-2" > )< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
G = < span class = "org-rainbow-delimiters-depth-1" > [< / span > G_ux_yx, G_uy_yx;
G_ux_yy, G_uy_yy< span class = "org-rainbow-delimiters-depth-1" > ]< / span > ;
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > save< span class = "org-rainbow-delimiters-depth-1" > (< / span > < span class = "org-string" > 'mat/plant.mat', 'G'< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
< / pre >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-org5cfd6fc" class = "outline-2" >
< h2 id = "org5cfd6fc" > < span class = "section-number-2" > 3< / span > Sensor Noise< / h2 >
< / div >
< div id = "outline-container-org2bd416c" class = "outline-2" >
< h2 id = "org2bd416c" > < span class = "section-number-2" > 4< / span > Plant Analysis< / h2 >
< div class = "outline-text-2" id = "text-4" >
< / div >
< div id = "outline-container-orge74c87e" class = "outline-3" >
< h3 id = "orge74c87e" > < span class = "section-number-3" > 4.1< / span > Rotation Matrix< / h3 >
< div class = "outline-text-3" id = "text-4-1" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > G0 = freqresp< span class = "org-rainbow-delimiters-depth-1" > (< / span > G, < span class = "org-highlight-numbers-number" > 0< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
< / pre >
< / div >
< / div >
< / div >
< / div >
< div id = "outline-container-org7b9c8d0" class = "outline-2" >
< h2 id = "org7b9c8d0" > < span class = "section-number-2" > 5< / span > Control Objective< / h2 >
< div class = "outline-text-2" id = "text-5" >
< p >
The maximum expected stroke is \(y_\text{max} = 3mm \approx 5e^{-2} rad\) at \(1Hz\).
The maximum wanted error is \(e_\text{max} = 10 \mu rad\).
< / p >
< p >
Thus, we require the sensitivity function at \(\omega_0 = 1\text{ Hz}\):
< / p >
\begin{align*}
|S(j\omega_0)| & < \left| \frac{e_\text{max}}{y_\text{max}} \right| \\
& < 2 \ cdot 10 ^ { -4 }
\end{align*}
< p >
In terms of loop gain, this is equivalent to:
\[ |L(j\omega_0)| > 5 \cdot 10^{3} \]
< / p >
< / div >
< / div >
< div id = "outline-container-orgbefa210" class = "outline-2" >
< h2 id = "orgbefa210" > < span class = "section-number-2" > 6< / span > Plant Scaling< / h2 >
< div class = "outline-text-2" id = "text-6" >
< ul class = "org-ul" >
< li > measured noise< / li >
< li > expected perturbations< / li >
< li > maximum input usage< / li >
< li > maximum wanted error< / li >
< / ul >
< / div >
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< / div >
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< div id = "outline-container-org90e1042" class = "outline-2" >
< h2 id = "org90e1042" > < span class = "section-number-2" > 7< / span > Control Design< / h2 >
2019-09-10 18:16:16 +02:00
< / div >
< / div >
< div id = "postamble" class = "status" >
< p class = "author" > Author: Dehaeze Thomas< / p >
2019-09-12 15:50:31 +02:00
< p class = "date" > Created: 2019-09-12 jeu. 15:20< / p >
2019-09-10 18:16:16 +02:00
< p class = "validation" > < a href = "http://validator.w3.org/check?uri=referer" > Validate< / a > < / p >
< / div >
< / body >
< / html >
