Add CSS/JS. Add plant identification scripts

This commit is contained in:
Thomas Dehaeze 2019-09-10 18:16:16 +02:00
parent 595d6c5188
commit c2607b8218
15 changed files with 2302 additions and 26 deletions

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<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>
<li><a href="#org25712dd">1. Identification</a>
<ul>
<li><a href="#orgc595eec">1.1. Excitation Data</a></li>
<li><a href="#orgd0be9c1">1.2. Input / Output data</a></li>
<li><a href="#org6ef1d0f">1.3. Estimation of the Frequency Response Function Matrix</a></li>
<li><a href="#orgb893c50">1.4. Coherence</a></li>
<li><a href="#org57d6ee3">1.5. Extraction of a transfer function matrix</a></li>
</ul>
</li>
<li><a href="#org6115789">2. Plant Analysis</a></li>
<li><a href="#orgcdff0c9">3. Control</a></li>
</ul>
</div>
</div>
<div id="outline-container-org25712dd" class="outline-2">
<h2 id="org25712dd"><span class="section-number-2">1</span> Identification</h2>
<div class="outline-text-2" id="text-1">
</div>
<div id="outline-container-orgc595eec" class="outline-3">
<h3 id="orgc595eec"><span class="section-number-3">1.1</span> Excitation Data</h3>
<div class="outline-text-3" id="text-1-1">
<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>
<div id="outline-container-orgd0be9c1" class="outline-3">
<h3 id="orgd0be9c1"><span class="section-number-3">1.2</span> Input / Output data</h3>
<div class="outline-text-3" id="text-1-2">
<p>
The identification data is loaded
</p>
<div class="org-src-container">
<pre class="src src-matlab">ux = load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'mat/data_001.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_002.mat', 't', 'uy', 'yx', 'yy'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
</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>
<div id="orgdd27b44" class="figure">
<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>
<div id="orgc1aa49f" class="figure">
<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>
<div id="outline-container-org6ef1d0f" class="outline-3">
<h3 id="org6ef1d0f"><span class="section-number-3">1.3</span> Estimation of the Frequency Response Function Matrix</h3>
<div class="outline-text-3" id="text-1-3">
<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>
<div id="org26ecff4" class="figure">
<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>
<div id="outline-container-orgb893c50" class="outline-3">
<h3 id="orgb893c50"><span class="section-number-3">1.4</span> Coherence</h3>
<div class="outline-text-3" id="text-1-4">
<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>
<div id="org2a5d4ff" class="figure">
<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>
<div id="outline-container-org57d6ee3" class="outline-3">
<h3 id="org57d6ee3"><span class="section-number-3">1.5</span> Extraction of a transfer function matrix</h3>
<div class="outline-text-3" id="text-1-5">
<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">&gt;</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">&amp;</span> omega<span class="org-type">&lt;</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">&gt;</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>
<div id="orgd766a83" class="figure">
<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>
<div id="orgc6407a4" class="figure">
<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>
<div id="outline-container-org6115789" class="outline-2">
<h2 id="org6115789"><span class="section-number-2">2</span> Plant Analysis</h2>
</div>
<div id="outline-container-orgcdff0c9" class="outline-2">
<h2 id="orgcdff0c9"><span class="section-number-2">3</span> Control</h2>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2019-09-10 mar. 18:15</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
</div>
</body>
</html>

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@ -6,6 +6,14 @@
#+EMAIL: dehaeze.thomas@gmail.com #+EMAIL: dehaeze.thomas@gmail.com
#+AUTHOR: Dehaeze Thomas #+AUTHOR: Dehaeze Thomas
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/htmlize.css"/>
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/readtheorg.css"/>
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/zenburn.css"/>
#+HTML_HEAD: <script type="text/javascript" src="./js/jquery.min.js"></script>
#+HTML_HEAD: <script type="text/javascript" src="./js/bootstrap.min.js"></script>
#+HTML_HEAD: <script type="text/javascript" src="./js/jquery.stickytableheaders.min.js"></script>
#+HTML_HEAD: <script type="text/javascript" src="./js/readtheorg.js"></script>
#+PROPERTY: header-args:matlab :session *MATLAB* #+PROPERTY: header-args:matlab :session *MATLAB*
#+PROPERTY: header-args:matlab+ :comments org #+PROPERTY: header-args:matlab+ :comments org
#+PROPERTY: header-args:matlab+ :results none #+PROPERTY: header-args:matlab+ :results none
@ -17,7 +25,8 @@
:END: :END:
* Matlab Init :noexport:ignore: * Identification
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir>> <<matlab-dir>>
#+end_src #+end_src
@ -26,7 +35,7 @@
<<matlab-init>> <<matlab-init>>
#+end_src #+end_src
* Identification of the Plant ** Excitation Data
#+begin_src matlab #+begin_src matlab
fs = 1e4; fs = 1e4;
Ts = 1/fs; Ts = 1/fs;
@ -56,35 +65,377 @@ Sample time: 0.0001 seconds
Discrete-time zero/pole/gain model. Discrete-time zero/pole/gain model.
#+end_example #+end_example
** Input / Output data
The identification data is loaded
#+begin_src matlab #+begin_src matlab
load('mat/data_001.mat', 't', 'ux') ux = load('mat/data_001.mat', 't', 'ux', 'yx', 'yy');
uy = load('mat/data_002.mat', 't', 'uy', 'yx', 'yy');
#+end_src
We remove the first seconds where the Cercalo is turned on.
#+begin_src matlab
i0x = 20*fs;
i0y = 10*fs;
ux.t = ux.t( i0x:end) - ux.t(i0x);
ux.ux = ux.ux(i0x:end);
ux.yx = ux.yx(i0x:end);
ux.yy = ux.yy(i0x:end);
uy.t = uy.t( i0y:end) - uy.t(i0x);
uy.uy = uy.uy(i0y:end);
uy.yx = uy.yx(i0y:end);
uy.yy = uy.yy(i0y:end);
#+end_src #+end_src
#+begin_src matlab #+begin_src matlab
figure; plot(t, ux) ux.ux = ux.ux-mean(ux.ux);
ux.yx = ux.yx-mean(ux.yx);
ux.yy = ux.yy-mean(ux.yy);
uy.ux = uy.ux-mean(uy.ux);
uy.yx = uy.yx-mean(uy.yx);
uy.yy = uy.yy-mean(uy.yy);
#+end_src #+end_src
#+begin_src matlab #+begin_src matlab :exports none
[tf_est, freqs] = tfestimate(ux, yx, hanning(ceil(length(ux)/10)), [], [], fs);
% h = idfrd(tf_est, freqs*2*pi, Ts);
#+end_src
#+begin_src matlab
figure; figure;
ax1 = subplot(2, 1, 1); ax1 = subplot(1, 2, 1);
plot(ux.t, ux.ux);
xlabel('Time [s]');
ylabel('Amplitude [V]');
legend({'$u_x$'});
ax2 = subplot(1, 2, 2);
hold on; hold on;
plot(freqs, abs(tf_est),'--') plot(ux.t, ux.yx, 'DisplayName', '$y_x$');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); plot(ux.t, ux.yy, 'DisplayName', '$y_y$');
ylabel('Amplitude [m/N]'); hold off;
xlabel('Time [s]');
ylabel('Amplitude [V]');
legend()
linkaxes([ax1,ax2],'x');
xlim([ux.t(1), ux.t(end)])
#+end_src
#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/identification_ux.pdf" :var figsize="full-normal" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+NAME: fig:identification_ux
#+CAPTION: Identification signals when exciting the $x$ axis ([[./figs/identification_ux.png][png]], [[./figs/identification_ux.pdf][pdf]])
[[file:figs/identification_ux.png]]
#+begin_src matlab :exports none
figure;
ax1 = subplot(1, 2, 1);
plot(uy.t, uy.uy);
xlabel('Time [s]');
ylabel('Amplitude [V]');
legend({'$u_y$'});
ax2 = subplot(1, 2, 2);
hold on;
plot(uy.t, uy.yy, 'DisplayName', '$y_y$');
plot(uy.t, uy.yx, 'DisplayName', '$y_x$');
hold off;
xlabel('Time [s]');
ylabel('Amplitude [V]');
legend()
linkaxes([ax1,ax2],'x');
xlim([uy.t(1), uy.t(end)])
#+end_src
#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/identification_uy.pdf" :var figsize="full-normal" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+NAME: fig:identification_uy
#+CAPTION: Identification signals when exciting the $y$ axis ([[./figs/identification_uy.png][png]], [[./figs/identification_uy.pdf][pdf]])
[[file:figs/identification_uy.png]]
** Estimation of the Frequency Response Function Matrix
We compute an estimate of the transfer functions.
#+begin_src matlab
[tf_ux_yx, f] = tfestimate(ux.ux, ux.yx, hanning(ceil(1*fs)), [], [], fs);
[tf_ux_yy, ~] = tfestimate(ux.ux, ux.yy, hanning(ceil(1*fs)), [], [], fs);
[tf_uy_yx, ~] = tfestimate(uy.uy, uy.yx, hanning(ceil(1*fs)), [], [], fs);
[tf_uy_yy, ~] = tfestimate(uy.uy, uy.yy, hanning(ceil(1*fs)), [], [], fs);
#+end_src
#+begin_src matlab :exports none
figure;
ax11 = subplot(2, 2, 1);
hold on;
plot(f, abs(tf_ux_yx))
title('Frequency Response Function $\frac{y_x}{u_x}$')
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
ylabel('Amplitude')
hold off; hold off;
ax2 = subplot(2, 1, 2); ax12 = subplot(2, 2, 2);
hold on; hold on;
plot(freqs, mod(180+180/pi*phase(tf_est), 360)-180,'--') plot(f, abs(tf_ux_yy))
set(gca,'xscale','log'); title('Frequency Response Function $\frac{y_x}{u_y}$')
ylim([-180, 180]); set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
yticks([-180, -90, 0, 90, 180]);
xlabel('Frequency [$Hz$]'); ylabel('Phase [deg]');
hold off; hold off;
ax21 = subplot(2, 2, 3);
hold on;
plot(f, abs(tf_uy_yx))
title('Frequency Response Function $\frac{y_y}{u_x}$')
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
ylabel('Amplitude')
xlabel('Frequency [Hz]')
hold off;
ax22 = subplot(2, 2, 4);
hold on;
plot(f, abs(tf_uy_yy))
title('Frequency Response Function $\frac{y_y}{u_y}$')
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
xlabel('Frequency [Hz]')
hold off;
linkaxes([ax11,ax12,ax21,ax22],'x');
xlim([10, 1000]);
linkaxes([ax11,ax12,ax21,ax22],'y');
ylim([1e-2, 1e3])
#+end_src #+end_src
#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/frequency_response_matrix.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+NAME: fig:frequency_response_matrix
#+CAPTION: Frequency Response Matrix ([[./figs/frequency_response_matrix.png][png]], [[./figs/frequency_response_matrix.pdf][pdf]])
[[file:figs/frequency_response_matrix.png]]
** Coherence
#+begin_src matlab
[coh_ux_yx, f] = mscohere(ux.ux, ux.yx, hanning(ceil(1*fs)), [], [], fs);
[coh_ux_yy, ~] = mscohere(ux.ux, ux.yy, hanning(ceil(1*fs)), [], [], fs);
[coh_uy_yx, ~] = mscohere(uy.uy, uy.yx, hanning(ceil(1*fs)), [], [], fs);
[coh_uy_yy, ~] = mscohere(uy.uy, uy.yy, hanning(ceil(1*fs)), [], [], fs);
#+end_src
#+begin_src matlab :exports none
figure;
ax11 = subplot(2, 2, 1);
hold on;
plot(f, coh_ux_yx)
set(gca, 'Xscale', 'log');
title('Coherence $\frac{y_x}{u_x}$')
ylabel('Coherence')
hold off;
ax12 = subplot(2, 2, 2);
hold on;
plot(f, coh_ux_yy)
set(gca, 'Xscale', 'log');
title('Coherence $\frac{y_x}{u_y}$')
hold off;
ax21 = subplot(2, 2, 3);
hold on;
plot(f, coh_uy_yx)
set(gca, 'Xscale', 'log');
title('Coherence $\frac{y_y}{u_x}$')
ylabel('Coherence')
xlabel('Frequency [Hz]')
hold off;
ax22 = subplot(2, 2, 4);
hold on;
plot(f, coh_uy_yy)
set(gca, 'Xscale', 'log');
title('Coherence $\frac{y_y}{u_y}$')
xlabel('Frequency [Hz]')
hold off;
linkaxes([ax11,ax12,ax21,ax22],'x');
xlim([10, 1000]);
linkaxes([ax11,ax12,ax21,ax22],'y');
ylim([0, 1])
#+end_src
#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/identification_coherence.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+NAME: fig:identification_coherence
#+CAPTION: Coherence ([[./figs/identification_coherence.png][png]], [[./figs/identification_coherence.pdf][pdf]])
[[file:figs/identification_coherence.png]]
** Extraction of a transfer function matrix
First we define the initial guess for the resonance frequencies and the weights associated.
#+begin_src matlab
freqs_res = [410, 250]; % [Hz]
freqs_res_weights = [10, 10]; % [Hz]
#+end_src
From the number of resonance frequency we want to fit, we define the order =N= of the system we want to obtain.
#+begin_src matlab
N = 2*length(freqs_res);
#+end_src
We then make an initial guess on the complex values of the poles.
#+begin_src matlab
xi = 0.001; % Approximate modal damping
poles = [2*pi*freqs_res*(xi + 1i), 2*pi*freqs_res*(xi - 1i)];
#+end_src
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.
#+begin_src matlab
weight = ones(1, length(f));
% weight = G_coh';
% alpha = 0.1;
% for freq_i = 1:length(freqs_res)
% weight(f>(1-alpha)*freqs_res(freq_i) & omega<(1 + alpha)*2*pi*freqs_res(freq_i)) = freqs_res_weights(freq_i);
% end
#+end_src
Ignore data above some frequency.
#+begin_src matlab
weight(f>1000) = 0;
#+end_src
#+begin_src matlab :exports none
figure;
hold on;
plot(f, weight);
plot(freqs_res, ones(size(freqs_res)), 'rx');
hold off;
xlabel('Frequency [Hz]');
xlabel('Weight Amplitude');
set(gca, 'xscale', 'log');
xlim([f(1), f(end)]);
#+end_src
#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/weights.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+NAME: fig:weights
#+CAPTION: Weights amplitude ([[./figs/weights.png][png]], [[./figs/weights.pdf][pdf]])
[[file:figs/weights.png]]
When we set some options for =vfit3=.
#+begin_src matlab
opts = struct();
opts.stable = 1; % Enforce stable poles
opts.asymp = 1; % Force D matrix to be null
opts.relax = 1; % Use vector fitting with relaxed non-triviality constraint
opts.skip_pole = 0; % Do NOT skip pole identification
opts.skip_res = 0; % Do NOT skip identification of residues (C,D,E)
opts.cmplx_ss = 0; % Create real state space model with block diagonal A
opts.spy1 = 0; % No plotting for first stage of vector fitting
opts.spy2 = 0; % Create magnitude plot for fitting of f(s)
#+end_src
We define the number of iteration.
#+begin_src matlab
Niter = 5;
#+end_src
An we run the =vectfit3= algorithm.
#+begin_src matlab
for iter = 1:Niter
[SER_ux_yx, poles, ~, fit_ux_yx] = vectfit3(tf_ux_yx.', 1i*2*pi*f, poles, weight, opts);
end
for iter = 1:Niter
[SER_uy_yx, poles, ~, fit_uy_yx] = vectfit3(tf_uy_yx.', 1i*2*pi*f, poles, weight, opts);
end
for iter = 1:Niter
[SER_ux_yy, poles, ~, fit_ux_yy] = vectfit3(tf_ux_yy.', 1i*2*pi*f, poles, weight, opts);
end
for iter = 1:Niter
[SER_uy_yy, poles, ~, fit_uy_yy] = vectfit3(tf_uy_yy.', 1i*2*pi*f, poles, weight, opts);
end
#+end_src
#+begin_src matlab :exports none
figure;
ax11 = subplot(2, 2, 1);
hold on;
plot(f, abs(tf_ux_yx))
plot(f, abs(fit_ux_yx))
title('Frequency Response Function $\frac{y_x}{u_x}$')
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
ylabel('Amplitude')
hold off;
ax12 = subplot(2, 2, 2);
hold on;
plot(f, abs(tf_ux_yy))
plot(f, abs(fit_ux_yy))
title('Frequency Response Function $\frac{y_x}{u_y}$')
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
hold off;
ax21 = subplot(2, 2, 3);
hold on;
plot(f, abs(tf_uy_yx))
plot(f, abs(fit_uy_yx))
title('Frequency Response Function $\frac{y_y}{u_x}$')
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
ylabel('Amplitude')
xlabel('Frequency [Hz]')
hold off;
ax22 = subplot(2, 2, 4);
hold on;
plot(f, abs(tf_uy_yy))
plot(f, abs(fit_uy_yy))
title('Frequency Response Function $\frac{y_y}{u_y}$')
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
xlabel('Frequency [Hz]')
hold off;
linkaxes([ax11,ax12,ax21,ax22],'x');
xlim([10, 1000]);
linkaxes([ax11,ax12,ax21,ax22],'y');
ylim([1e-2, 1e3])
#+end_src
#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/identification_matrix_fit.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+NAME: fig:identification_matrix_fit
#+CAPTION: Transfer Function Extraction of the FRF matrix ([[./figs/identification_matrix_fit.png][png]], [[./figs/identification_matrix_fit.pdf][pdf]])
[[file:figs/identification_matrix_fit.png]]
And finally, we create the identified state space model:
#+begin_src matlab
G_ux_yx = minreal(ss(full(SER_ux_yx.A),SER_ux_yx.B,SER_ux_yx.C,SER_ux_yx.D));
G_uy_yx = minreal(ss(full(SER_uy_yx.A),SER_uy_yx.B,SER_uy_yx.C,SER_uy_yx.D));
G_ux_yy = minreal(ss(full(SER_ux_yy.A),SER_ux_yy.B,SER_ux_yy.C,SER_ux_yy.D));
G_uy_yy = minreal(ss(full(SER_uy_yy.A),SER_uy_yy.B,SER_uy_yy.C,SER_uy_yy.D));
G = [G_ux_yx, G_uy_yx;
G_ux_yy, G_uy_yy];
#+end_src
#+begin_src matlab
save('mat/plant.mat', 'G');
#+end_src
* Plant Analysis
* Control

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$(function() {
$('.note').before("<p class='admonition-title note'>Note</p>");
$('.seealso').before("<p class='admonition-title seealso'>See also</p>");
$('.warning').before("<p class='admonition-title warning'>Warning</p>");
$('.caution').before("<p class='admonition-title caution'>Caution</p>");
$('.attention').before("<p class='admonition-title attention'>Attention</p>");
$('.tip').before("<p class='admonition-title tip'>Tip</p>");
$('.important').before("<p class='admonition-title important'>Important</p>");
$('.hint').before("<p class='admonition-title hint'>Hint</p>");
$('.error').before("<p class='admonition-title error'>Error</p>");
$('.danger').before("<p class='admonition-title danger'>Danger</p>");
});
$( document ).ready(function() {
// Shift nav in mobile when clicking the menu.
$(document).on('click', "[data-toggle='wy-nav-top']", function() {
$("[data-toggle='wy-nav-shift']").toggleClass("shift");
$("[data-toggle='rst-versions']").toggleClass("shift");
});
// Close menu when you click a link.
$(document).on('click', ".wy-menu-vertical .current ul li a", function() {
$("[data-toggle='wy-nav-shift']").removeClass("shift");
$("[data-toggle='rst-versions']").toggleClass("shift");
});
$(document).on('click', "[data-toggle='rst-current-version']", function() {
$("[data-toggle='rst-versions']").toggleClass("shift-up");
});
// Make tables responsive
$("table.docutils:not(.field-list)").wrap("<div class='wy-table-responsive'></div>");
});
$( document ).ready(function() {
$('#text-table-of-contents ul').first().addClass('nav');
// ScrollSpy also requires that we use
// a Bootstrap nav component.
$('body').scrollspy({target: '#text-table-of-contents'});
// add sticky table headers
$('table').stickyTableHeaders();
// set the height of tableOfContents
var $postamble = $('#postamble');
var $tableOfContents = $('#table-of-contents');
$tableOfContents.css({paddingBottom: $postamble.outerHeight()});
// add TOC button
var toggleSidebar = $('<div id="toggle-sidebar"><a href="#table-of-contents"><h2>Table of Contents</h2></a></div>');
$('#content').prepend(toggleSidebar);
// add close button when sidebar showed in mobile screen
var closeBtn = $('<a class="close-sidebar" href="#">Close</a>');
var tocTitle = $('#table-of-contents').find('h2');
tocTitle.append(closeBtn);
});
window.SphinxRtdTheme = (function (jquery) {
var stickyNav = (function () {
var navBar,
win,
stickyNavCssClass = 'stickynav',
applyStickNav = function () {
if (navBar.height() <= win.height()) {
navBar.addClass(stickyNavCssClass);
} else {
navBar.removeClass(stickyNavCssClass);
}
},
enable = function () {
applyStickNav();
win.on('resize', applyStickNav);
},
init = function () {
navBar = jquery('nav.wy-nav-side:first');
win = jquery(window);
};
jquery(init);
return {
enable : enable
};
}());
return {
StickyNav : stickyNav
};
}($));

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