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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 = "#org56fbace" > 1. Introduction< / a >
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< ul >
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< li > < a href = "#orgc71c3a8" > 1.1. Block Diagram< / a > < / li >
< li > < a href = "#org6dd135b" > 1.2. Amplifier for the Cercalo< / a > < / li >
< li > < a href = "#org3f4c55a" > 1.3. Cercalo< / a > < / li >
< li > < a href = "#org8cf1cba" > 1.4. Optical Setup< / a > < / li >
< li > < a href = "#org6d8a72e" > 1.5. Newport< / a > < / li >
< li > < a href = "#orgcf9e532" > 1.6. 4 quadrant Diode< / a > < / li >
< li > < a href = "#orgbfa25c0" > 1.7. ADC/DAC< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org549f783" > 2. Identification< / a >
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< ul >
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< li > < a href = "#orga174838" > 2.1. Excitation Data< / a > < / li >
< li > < a href = "#org4707d35" > 2.2. Signals< / a > < / li >
< li > < a href = "#org96d4dae" > 2.3. Huddle Test< / a > < / li >
< li > < a href = "#org5b29539" > 2.4. Input / Output data< / a > < / li >
< li > < a href = "#org0556947" > 2.5. Estimation of the Frequency Response Function Matrix< / a > < / li >
< li > < a href = "#org6c025b6" > 2.6. Coherence< / a > < / li >
< li > < a href = "#org74eeb16" > 2.7. Extraction of a transfer function matrix< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orge157b63" > 3. Calibration of the 4 Quadrant Diode< / a > < / li >
< li > < a href = "#orgc276bb8" > 4. Plant Scaling< / a > < / li >
< li > < a href = "#orgffff224" > 5. Plant Analysis< / a >
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< ul >
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< li > < a href = "#orge9e5454" > 5.1. Load Plant< / a > < / li >
< li > < a href = "#org425e4e2" > 5.2. RGA-Number< / a > < / li >
< li > < a href = "#orge80bcc0" > 5.3. Rotation Matrix< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgfd824b0" > 6. Control Objective< / a > < / li >
< li > < a href = "#orgc7afa66" > 7. Control Design< / a > < / li >
< li > < a href = "#org5791267" > 8. Measurement of the non-repeatability< / a > < / li >
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< / ul >
< / div >
< / div >
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< div id = "outline-container-org56fbace" class = "outline-2" >
< h2 id = "org56fbace" > < span class = "section-number-2" > 1< / span > Introduction< / h2 >
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< div class = "outline-text-2" id = "text-1" >
< / div >
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< div id = "outline-container-orgc71c3a8" class = "outline-3" >
< h3 id = "orgc71c3a8" > < span class = "section-number-3" > 1.1< / span > Block Diagram< / h3 >
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< div class = "outline-text-3" id = "text-1-1" >
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< p >
The block diagram of the setup to be controlled is shown in Fig. < a href = "#orgc07f042" > 1< / a > .
< / p >
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< div id = "orgc07f042" class = "figure" >
< p > < img src = "figs/cercalo_diagram_simplify.png" alt = "cercalo_diagram_simplify.png" / >
< / p >
< p > < span class = "figure-number" > Figure 1: < / span > Block Diagram of the Experimental Setup< / p >
< / div >
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< p >
The transfer functions in the system are:
< / p >
< ul class = "org-ul" >
< li > < b > Current Amplifier< / b > : from the voltage set by the DAC to the voltage across the Cercalo inductors
\[ G_i = \begin{bmatrix} G_{i,h} & 0 \\ 0 & G_{i,v} \end{bmatrix} \]< / li >
< li > < b > Voltage Amplifier< / b > : from the voltage across the Cercalo inductors to the measured voltage
\[ G_a = \begin{bmatrix} G_{a,h} & 0 \\ 0 & G_{a,v} \end{bmatrix} \]< / li >
< li > < b > Cercalo< / b > : Transfer function from the Voltage across the cercalo inductors to the 4 quadrant measurement
\[ G_c = \begin{bmatrix} G_{\frac{V_{p,h}}{\tilde{U}_{c,h}}} & G_{\frac{V_{p,h}}{\tilde{U}_{c,v}}} \\ G_{\frac{V_{p,v}}{\tilde{U}_{c,h}}} & G_{\frac{V_{p,v}}{\tilde{U}_{c,v}}} \end{bmatrix} \]< / li >
< li > < b > Newport< / b > Transfer function from the command signal of the Newport to the 4 quadrant measurement
\[ G_n = \begin{bmatrix} G_{\frac{V_{p,h}}{U_{n,h}}} & G_{\frac{V_{p,h}}{U_{n,v}}} \\ G_{\frac{V_{p,v}}{U_{n,h}}} & G_{\frac{V_{n,v}}{U_{n,v}}} \end{bmatrix} \]< / li >
< / ul >
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< p >
The block diagram with each transfer function is shown in Fig. < a href = "#orgc0fd79e" > 2< / a > .
< / p >
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< div id = "orgc0fd79e" class = "figure" >
< p > < img src = "figs/cercalo_diagram.png" alt = "cercalo_diagram.png" / >
< / p >
< p > < span class = "figure-number" > Figure 2: < / span > Block Diagram of the Experimental Setup with detailed dynamics< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-org6dd135b" class = "outline-3" >
< h3 id = "org6dd135b" > < span class = "section-number-3" > 1.2< / span > Amplifier for the Cercalo< / h3 >
< div class = "outline-text-3" id = "text-1-2" >
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< div class = "figure" >
< p > < img src = "figs/cercalo_amplifier.png" alt = "cercalo_amplifier.png" / >
< / p >
< / div >
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< p >
The value of the resistor in series with the buffer have been measured for both axis.
< / p >
< ul class = "org-ul" >
< li > \(R_h = 41 \Omega\)< / li >
< li > \(L_{c,h} = 0.1 mH\)< / li >
< li > \(R_{c,h} = 9.3 \Omega\)< / li >
< li > \(R_v = 41 \Omega\)< / li >
< li > \(L_{c,v} = 0.1 mH\)< / li >
< li > \(R_{c,v} = 8.3 \Omega\)< / li >
< / ul >
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< p >
We want to find the transfer function from \(U_c\) to \(V_L\) and from \(U_c\) to \(i_c\).
< / p >
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< p >
We have that:
< / p >
\begin{align*}
V_C & = R_c i + L_c s i \\
U_c & = (R + R_c) i + L_c s i
\end{align*}
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< p >
Thus:
< / p >
\begin{align}
\frac{i_c}{U_c} & = \frac{1}{(R + R_c) + L_c s} \\
& = \frac{G_0}{1 + s/\omega_0}
\end{align}
< p >
with
< / p >
< ul class = "org-ul" >
< li > \(G_{0,i} = \frac{1}{R + R_c}\)< / li >
< li > \(\omega_0 = \frac{R + R_c}{L_c}\)< / li >
< / ul >
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< p >
And
< / p >
\begin{align}
\frac{V_c}{U_c} & = \frac{R_c + L_c s}{(R + R_c) + L_c s} \\
& = \frac{\frac{R_c}{R + R_c} + \frac{L_c}{R + R_c} s}{1 + \frac{L_c}{R + R_c} s} \\
& = \frac{G_0 + s/\omega_0}{1 + s/\omega_0} \\
\end{align}
< p >
with
< / p >
< ul class = "org-ul" >
< li > \(G_0 = \frac{R_c}{R + R_c}\)< / li >
< li > \(\omega_0 = \frac{R + R_c}{L_c}\)< / li >
< / ul >
< p >
Let's verify that the electrical circuit behaves as a constant current amplifier in the frequency band of interest.
< / p >
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Gvh = < span class = "org-rainbow-delimiters-depth-1" > (< / span > Rch < span class = "org-type" > +< / span > Lch < span class = "org-type" > *< / span > s< span class = "org-rainbow-delimiters-depth-1" > )< / span > < span class = "org-type" > /< / span > < span class = "org-rainbow-delimiters-depth-1" > (< / span > Rh < span class = "org-type" > +< / span > Rch < span class = "org-type" > +< / span > Lch < span class = "org-type" > *< / span > s< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
Giv = < span class = "org-highlight-numbers-number" > 1< / span > < span class = "org-type" > /< / span > < span class = "org-rainbow-delimiters-depth-1" > (< / span > Rv < span class = "org-type" > +< / span > Rcv < span class = "org-type" > +< / span > Lcv < span class = "org-type" > *< / span > s< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
Gvv = < span class = "org-rainbow-delimiters-depth-1" > (< / span > Rcv < span class = "org-type" > +< / span > Lcv < span class = "org-type" > *< / span > s< span class = "org-rainbow-delimiters-depth-1" > )< / span > < span class = "org-type" > /< / span > < span class = "org-rainbow-delimiters-depth-1" > (< / span > Rv < span class = "org-type" > +< / span > Rcv < span class = "org-type" > +< / span > Lcv < span class = "org-type" > *< / span > s< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Gih0 = freqresp< span class = "org-rainbow-delimiters-depth-1" > (< / span > Gih, < span class = "org-highlight-numbers-number" > 0< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
Gvh0 = freqresp< span class = "org-rainbow-delimiters-depth-1" > (< / span > Gvh, < span class = "org-highlight-numbers-number" > 0< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
Giv0 = freqresp< span class = "org-rainbow-delimiters-depth-1" > (< / span > Giv, < span class = "org-highlight-numbers-number" > 0< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
Gvv0 = freqresp< span class = "org-rainbow-delimiters-depth-1" > (< / span > Gvv, < span class = "org-highlight-numbers-number" > 0< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
< / pre >
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< / div >
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< div id = "orgfec4858" class = "figure" >
< p > < img src = "figs/current_amplifier_tf.png" alt = "current_amplifier_tf.png" / >
< / p >
< p > < span class = "figure-number" > Figure 4: < / span > Transfer function for the current amplifier (< a href = "./figs/current_amplifier_tf.png" > png< / a > , < a href = "./figs/current_amplifier_tf.pdf" > pdf< / a > )< / p >
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< / div >
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< div class = "important" >
< p >
The current amplifier has a constant gain over all the frequency band of interest.
\[ G_i(s) = \begin{bmatrix} 0.02 & 0 \\ 0 & 0.02 \end{bmatrix}\quad \left[\frac{A}{V}\right] \]
\[ G_a(s) = \begin{bmatrix} 0.185 & 0 \\ 0 & 0.168 \end{bmatrix} \left[\frac{V}{V}\right] \]
< / p >
< / div >
< / div >
< / div >
< div id = "outline-container-org3f4c55a" class = "outline-3" >
< h3 id = "org3f4c55a" > < span class = "section-number-3" > 1.3< / span > Cercalo< / h3 >
< div class = "outline-text-3" id = "text-1-3" >
< p >
From the Cercalo documentation, we have the parameters shown on table < a href = "#orgfb4a003" > 1< / a > .
< / p >
< table id = "orgfb4a003" border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
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< caption class = "t-above" > < span class = "table-number" > Table 1:< / span > Cercalo Parameters< / caption >
< colgroup >
< col class = "org-left" / >
< col class = "org-right" / >
< col class = "org-right" / >
< col class = "org-right" / >
< col class = "org-right" / >
< col class = "org-right" / >
< / colgroup >
< thead >
< tr >
< th scope = "col" class = "org-left" >   < / th >
< th scope = "col" class = "org-right" > Maximum Stroke [deg]< / th >
< th scope = "col" class = "org-right" > Resonance Frequency [Hz]< / th >
< th scope = "col" class = "org-right" > DC Gain [mA/deg]< / th >
< th scope = "col" class = "org-right" > Gain at resonance [deg/V]< / th >
< th scope = "col" class = "org-right" > RC Resistance [Ohm]< / th >
< / tr >
< / thead >
< tbody >
< tr >
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< td class = "org-left" > AX1 (Horizontal)< / td >
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< td class = "org-right" > 5< / td >
< td class = "org-right" > 411.13< / td >
< td class = "org-right" > 28.4< / td >
< td class = "org-right" > 382.9< / td >
< td class = "org-right" > 9.41< / td >
< / tr >
< tr >
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< td class = "org-left" > AX2 (Vertical)< / td >
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< td class = "org-right" > 5< / td >
< td class = "org-right" > 252.5< / td >
< td class = "org-right" > 35.2< / td >
< td class = "org-right" > 350.4< / td >
< td class = "org-right" >   < / td >
< / tr >
< / tbody >
< / table >
< p >
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The Inductance and DC resistance of the two axis of the Cercalo have been measured:
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< / p >
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< ul class = "org-ul" >
< li > \(L_{c,h} = 0.1\ \text{mH}\)< / li >
< li > \(L_{c,v} = 0.1\ \text{mH}\)< / li >
< li > \(R_{c,h} = 9.3\ \Omega\)< / li >
< li > \(R_{c,v} = 8.3\ \Omega\)< / li >
< / ul >
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< p >
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Let's first consider the < b > horizontal direction< / b > and we try to model the Cercalo by a spring/mass/damper system (Fig. < a href = "#orgd829094" > 5< / a > ).
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< / p >
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< div id = "orgd829094" class = "figure" >
< p > < img src = "figs/mech_cercalo.png" alt = "mech_cercalo.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 5: < / span > 1 degree-of-freedom model of the Cercalo< / p >
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< / div >
< p >
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The equation of motion is:
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< / p >
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\begin{align*}
\frac{x}{F} & = \frac{1}{k + c s + m s^2} \\
& = \frac{G_0}{1 + 2 \xi \frac{s}{\omega_0} + \frac{s^2}{\omega_0^2}}
\end{align*}
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< p >
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with:
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< / p >
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< ul class = "org-ul" >
< li > \(G_0 = 1/k\) is the gain at DC in rad/N< / li >
< li > \(\xi = \frac{c}{2 \sqrt{km}}\) is the damping ratio of the system< / li >
< li > \(\omega_0 = \sqrt{\frac{k}{m}}\) is the resonance frequency in rad< / li >
< / ul >
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< p >
The force \(F\) applied to the mass is proportional to the current \(I\) flowing through the voice coils:
\[ \frac{F}{I} = \alpha \]
with \(\alpha\) is in \(N/A\) and is to be determined.
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< / p >
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< p >
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The current \(I\) is also proportional to the voltage at the output of the buffer:
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< / p >
\begin{align*}
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\frac{I_c}{U_c} & = \frac{1}{(R + R_c) + L_c s} \\
& \approx 0.02 \left[ \frac{A}{V} \right]
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\end{align*}
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< p >
Let's try to determine the equivalent mass and spring values.
From table < a href = "#orgfb4a003" > 1< / a > , for the horizontal direction:
\[ \left| \frac{x}{I} \right|(0) = \left| \alpha \frac{x}{F} \right|(0) = 28.4\ \frac{mA}{deg} = 1.63\ \frac{A}{rad} \]
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< / p >
< p >
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So:
\[ \alpha \frac{1}{k} = 1.63 \Longleftrightarrow k = \frac{\alpha}{1.63} \left[\frac{N}{rad}\right] \]
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< / p >
< p >
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We also know the resonance frequency:
\[ \omega_0 = 411.1\ \text{Hz} = 2583\ \frac{rad}{s} \]
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< / p >
< p >
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And the gain at resonance:
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< / p >
\begin{align*}
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\left| \frac{x}{U_c} \right|(j\omega_0) & = \left| 0.02 \frac{x}{I_c} \right| (j\omega_0) \\
& = \left| 0.02 \alpha \frac{x}{F} \right| (j\omega_0) \\
& = 0.02 \alpha \frac{1/k}{2\xi} \\
& = 282.9\ \left[\frac{deg}{V}\right] \\
& = 4.938\ \left[\frac{rad}{V}\right]
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\end{align*}
< p >
Thus:
< / p >
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\begin{align*}
& \frac{\alpha}{2 \xi k} = 245 \\
\Leftrightarrow & \frac{1.63}{2 \xi} = 245 \\
\Leftrightarrow & \xi = 0.0033 \\
\Leftrightarrow & \xi = 0.33 \%
\end{align*}
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< div class = "important" >
\begin{align*}
G_0 & = \frac{1.63}{\alpha}\ \frac{rad}{N} \\
\xi & = 0.0033 \\
\omega_0 & = 2583\ \frac{rad}{s}
\end{align*}
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< p >
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and in terms of the physical properties:
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< / p >
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\begin{align*}
k & = \frac{\alpha}{1.63}\ \frac{N}{rad} \\
\xi & = 0.0033 \\
m & = \frac{\alpha}{1.1 \cdot 10^7}\ \frac{kg}{m^2}
\end{align*}
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< p >
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Thus, we have to determine \(\alpha\).
This can be done experimentally by determining the gain at DC or at resonance of the system.
For that, we need to know the angle of the mirror, thus we need to < b > calibrate< / b > the photo-diodes.
This will be done using the Newport.
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< / p >
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< / div >
< / div >
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< / div >
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< div id = "outline-container-org8cf1cba" class = "outline-3" >
< h3 id = "org8cf1cba" > < span class = "section-number-3" > 1.4< / span > Optical Setup< / h3 >
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< / div >
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< div id = "outline-container-org6d8a72e" class = "outline-3" >
< h3 id = "org6d8a72e" > < span class = "section-number-3" > 1.5< / span > Newport< / h3 >
< div class = "outline-text-3" id = "text-1-5" >
< p >
Parameters of the Newport are shown in Fig. < a href = "#org893d45e" > 6< / a > .
< / p >
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< p >
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It's dynamics for small angle excitation is shown in Fig. < a href = "#org4e41b71" > 7< / a > .
< / p >
< p >
And we have:
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< / p >
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\begin{align*}
G_{n, h}(0) & = 2.62 \cdot 10^{-3}\ \frac{rad}{V} \\
G_{n, v}(0) & = 2.62 \cdot 10^{-3}\ \frac{rad}{V}
\end{align*}
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< div id = "org893d45e" class = "figure" >
< p > < img src = "figs/newport_doc.png" alt = "newport_doc.png" / >
< / p >
< p > < span class = "figure-number" > Figure 6: < / span > Documentation of the Newport< / p >
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< / div >
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< div id = "org4e41b71" class = "figure" >
< p > < img src = "figs/newport_gain.png" alt = "newport_gain.png" / >
< / p >
< p > < span class = "figure-number" > Figure 7: < / span > Transfer function of the Newport< / p >
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< / div >
< / div >
< / div >
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< div id = "outline-container-orgcf9e532" class = "outline-3" >
< h3 id = "orgcf9e532" > < span class = "section-number-3" > 1.6< / span > 4 quadrant Diode< / h3 >
< div class = "outline-text-3" id = "text-1-6" >
< p >
The front view of the 4 quadrant photo-diode is shown in Fig. < a href = "#org279009d" > 8< / a > .
< / p >
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< div id = "org279009d" class = "figure" >
< p > < img src = "figs/4qd_naming.png" alt = "4qd_naming.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 8: < / span > Front view of the 4QD< / p >
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< / div >
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< p >
Each of the photo-diode is amplified using a 4-channel amplifier as shown in Fig. < a href = "#org7f931ef" > 9< / a > .
< / p >
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< div id = "org7f931ef" class = "figure" >
< p > < img src = "figs/4qd_amplifier.png" alt = "4qd_amplifier.png" / >
< / p >
< p > < span class = "figure-number" > Figure 9: < / span > Wiring of the amplifier. The amplifier is located on the bottom right of the board< / p >
< / div >
< / div >
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< / div >
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< div id = "outline-container-orgbfa25c0" class = "outline-3" >
< h3 id = "orgbfa25c0" > < span class = "section-number-3" > 1.7< / span > ADC/DAC< / h3 >
< div class = "outline-text-3" id = "text-1-7" >
< p >
Let's compute the theoretical noise of the ADC/DAC.
< / p >
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\begin{align*}
\Delta V & = 20 V \\
n & = 16bits \\
q & = \Delta V/2^n = 305 \mu V \\
f_N & = 10kHz \\
\Gamma_n & = \frac{q^2}{12 f_N} = 7.76 \cdot 10^{-13} \frac{V^2}{Hz}
\end{align*}
< p >
with \(\Delta V\) the total range of the ADC, \(n\) its number of bits, \(q\) the quantization, \(f_N\) the sampling frequency and \(\Gamma_n\) its theoretical Power Spectral Density.
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< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-org549f783" class = "outline-2" >
< h2 id = "org549f783" > < span class = "section-number-2" > 2< / span > Identification< / h2 >
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< div class = "outline-text-2" id = "text-2" >
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< p >
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< a id = "orgaf96e37" > < / a >
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< / 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-orga174838" class = "outline-3" >
< h3 id = "orga174838" > < span class = "section-number-3" > 2.1< / span > Excitation Data< / h3 >
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< 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-org4707d35" class = "outline-3" >
< h3 id = "org4707d35" > < span class = "section-number-3" > 2.2< / span > Signals< / h3 >
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< div class = "outline-text-3" id = "text-2-2" >
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< table border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
< colgroup >
< col class = "org-left" / >
< col class = "org-left" / >
< col class = "org-left" / >
< / colgroup >
< thead >
< tr >
< th scope = "col" class = "org-left" > Signal< / th >
< th scope = "col" class = "org-left" > Name< / th >
< th scope = "col" class = "org-left" > Unit< / th >
< / tr >
< / thead >
< tbody >
< tr >
< td class = "org-left" > Voltage Sent to Cercalo - Horizontal< / td >
< td class = "org-left" > < code > Uch< / code > < / td >
< td class = "org-left" > [V]< / td >
< / tr >
< tr >
< td class = "org-left" > Voltage Sent to Cercalo - Vertical< / td >
< td class = "org-left" > < code > Ucv< / code > < / td >
< td class = "org-left" > [V]< / td >
< / tr >
< tr >
< td class = "org-left" > Voltage Sent to Newport - Horizontal< / td >
< td class = "org-left" > < code > Unh< / code > < / td >
< td class = "org-left" > [V]< / td >
< / tr >
< tr >
< td class = "org-left" > Voltage Sent to Newport - Vertical< / td >
< td class = "org-left" > < code > Unv< / code > < / td >
< td class = "org-left" > [V]< / td >
< / tr >
< / tbody >
< tbody >
< tr >
< td class = "org-left" > 4Q Photodiode Measurement - Horizontal< / td >
< td class = "org-left" > < code > Vph< / code > < / td >
< td class = "org-left" > [V]< / td >
< / tr >
< tr >
< td class = "org-left" > 4Q Photodiode Measurement - Vertical< / td >
< td class = "org-left" > < code > Vpv< / code > < / td >
< td class = "org-left" > [V]< / td >
< / tr >
< tr >
< td class = "org-left" > Measured Voltage across the Inductance - Horizontal< / td >
< td class = "org-left" > < code > Vch< / code > < / td >
< td class = "org-left" > [V]< / td >
< / tr >
< tr >
< td class = "org-left" > Measured Voltage across the Inductance - Vertical< / td >
< td class = "org-left" > < code > Vcv< / code > < / td >
< td class = "org-left" > [V]< / td >
< / tr >
< tr >
< td class = "org-left" > Newport Metrology - Horizontal< / td >
< td class = "org-left" > < code > Vnh< / code > < / td >
< td class = "org-left" > [V]< / td >
< / tr >
< tr >
< td class = "org-left" > Newport Metrology - Vertical< / td >
< td class = "org-left" > < code > Vnv< / code > < / td >
< td class = "org-left" > [V]< / td >
< / tr >
< / tbody >
< tbody >
< tr >
< td class = "org-left" > Attocube Measurement< / td >
< td class = "org-left" > < code > Va< / code > < / td >
< td class = "org-left" > [m]< / td >
< / tr >
< / tbody >
< / table >
< / div >
< / div >
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< div id = "outline-container-org96d4dae" class = "outline-3" >
< h3 id = "org96d4dae" > < span class = "section-number-3" > 2.3< / span > Huddle Test< / h3 >
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< div class = "outline-text-3" id = "text-2-3" >
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< p >
We load the data taken during the Huddle Test.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > load< span class = "org-rainbow-delimiters-depth-1" > (< / span > < span class = "org-string" > 'mat/data_test.mat'< / span > , < span class = "org-underline" > ...< / span >
< span class = "org-string" > 't', 'Uch', 'Ucv'< / span > , < span class = "org-underline" > ...< / span >
< span class = "org-string" > 'Unh', 'Unv'< / span > , < span class = "org-underline" > ...< / span >
< span class = "org-string" > 'Vph', 'Vpv'< / span > , < span class = "org-underline" > ...< / span >
< span class = "org-string" > 'Vch', 'Vcv'< / span > , < span class = "org-underline" > ...< / span >
< span class = "org-string" > 'Vnh', 'Vnv'< / span > , < span class = "org-underline" > ...< / span >
< span class = "org-string" > 'Va'< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
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< / pre >
< / div >
< p >
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We remove the first second of data where everything is settling down.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > t0 = < span class = "org-highlight-numbers-number" > 1< / span > ;
Uch< span class = "org-rainbow-delimiters-depth-1" > (< / span > t< span class = "org-type" > < < / span > t0< span class = "org-rainbow-delimiters-depth-1" > )< / span > = < span class = "org-rainbow-delimiters-depth-1" > []< / span > ;
Ucv< span class = "org-rainbow-delimiters-depth-1" > (< / span > t< span class = "org-type" > < < / span > t0< span class = "org-rainbow-delimiters-depth-1" > )< / span > = < span class = "org-rainbow-delimiters-depth-1" > []< / span > ;
Unh< span class = "org-rainbow-delimiters-depth-1" > (< / span > t< span class = "org-type" > < < / span > t0< span class = "org-rainbow-delimiters-depth-1" > )< / span > = < span class = "org-rainbow-delimiters-depth-1" > []< / span > ;
Unv< span class = "org-rainbow-delimiters-depth-1" > (< / span > t< span class = "org-type" > < < / span > t0< span class = "org-rainbow-delimiters-depth-1" > )< / span > = < span class = "org-rainbow-delimiters-depth-1" > []< / span > ;
Vph< span class = "org-rainbow-delimiters-depth-1" > (< / span > t< span class = "org-type" > < < / span > t0< span class = "org-rainbow-delimiters-depth-1" > )< / span > = < span class = "org-rainbow-delimiters-depth-1" > []< / span > ;
Vpv< span class = "org-rainbow-delimiters-depth-1" > (< / span > t< span class = "org-type" > < < / span > t0< span class = "org-rainbow-delimiters-depth-1" > )< / span > = < span class = "org-rainbow-delimiters-depth-1" > []< / span > ;
Vch< span class = "org-rainbow-delimiters-depth-1" > (< / span > t< span class = "org-type" > < < / span > t0< span class = "org-rainbow-delimiters-depth-1" > )< / span > = < span class = "org-rainbow-delimiters-depth-1" > []< / span > ;
Vcv< span class = "org-rainbow-delimiters-depth-1" > (< / span > t< span class = "org-type" > < < / span > t0< span class = "org-rainbow-delimiters-depth-1" > )< / span > = < span class = "org-rainbow-delimiters-depth-1" > []< / span > ;
Vnh< span class = "org-rainbow-delimiters-depth-1" > (< / span > t< span class = "org-type" > < < / span > t0< span class = "org-rainbow-delimiters-depth-1" > )< / span > = < span class = "org-rainbow-delimiters-depth-1" > []< / span > ;
Vnv< span class = "org-rainbow-delimiters-depth-1" > (< / span > t< span class = "org-type" > < < / span > t0< span class = "org-rainbow-delimiters-depth-1" > )< / span > = < span class = "org-rainbow-delimiters-depth-1" > []< / span > ;
Va< span class = "org-rainbow-delimiters-depth-1" > (< / span > t< span class = "org-type" > < < / span > t0< 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 > t0< 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 > ; % We start at t=< span class = "org-highlight-numbers-number" > 0< / span >
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< / pre >
< / div >
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< p >
We compute the Power Spectral Density of the horizontal and vertical positions of the beam as measured by the 4 quadrant diode.
< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-rainbow-delimiters-depth-1" > [< / span > psd_Vph, f< span class = "org-rainbow-delimiters-depth-1" > ]< / span > = pwelch< span class = "org-rainbow-delimiters-depth-1" > (< / span > Vph, 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_Vpv, < span class = "org-type" > ~< / span > < span class = "org-rainbow-delimiters-depth-1" > ]< / span > = pwelch< span class = "org-rainbow-delimiters-depth-1" > (< / span > Vpv, 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 > ;
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< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-type" > figure< / span > ;
hold on;
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plot< span class = "org-rainbow-delimiters-depth-1" > (< / span > f, sqrt< span class = "org-rainbow-delimiters-depth-2" > (< / span > psd_Vph< 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 > Vp_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_Vpv< 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 > Vp_v< span class = "org-rainbow-delimiters-depth-2" > }< / span > $'< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
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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 >
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< p >
We compute the Power Spectral Density of the voltage across the inductance used for horizontal and vertical positioning of the Cercalo.
< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-rainbow-delimiters-depth-1" > [< / span > psd_Vch, f< span class = "org-rainbow-delimiters-depth-1" > ]< / span > = pwelch< span class = "org-rainbow-delimiters-depth-1" > (< / span > Vch, 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_Vcv, < span class = "org-type" > ~< / span > < span class = "org-rainbow-delimiters-depth-1" > ]< / span > = pwelch< span class = "org-rainbow-delimiters-depth-1" > (< / span > Vcv, 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 > ;
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< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-type" > figure< / span > ;
hold on;
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plot< span class = "org-rainbow-delimiters-depth-1" > (< / span > f, sqrt< span class = "org-rainbow-delimiters-depth-2" > (< / span > psd_Vch< 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 > Vc_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_Vcv< 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 > Vc_v< span class = "org-rainbow-delimiters-depth-2" > }< / span > $'< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
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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 >
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< div id = "outline-container-org5b29539" class = "outline-3" >
< h3 id = "org5b29539" > < span class = "section-number-3" > 2.4< / span > Input / Output data< / h3 >
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< div class = "outline-text-3" id = "text-2-4" >
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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" > uh = load< span class = "org-rainbow-delimiters-depth-1" > (< / span > < span class = "org-string" > 'mat/data_uh.mat'< / span > , < span class = "org-underline" > ...< / span >
< span class = "org-string" > 't', 'Uch', 'Ucv'< / span > , < span class = "org-underline" > ...< / span >
< span class = "org-string" > 'Unh', 'Unv'< / span > , < span class = "org-underline" > ...< / span >
< span class = "org-string" > 'Vph', 'Vpv'< / span > , < span class = "org-underline" > ...< / span >
< span class = "org-string" > 'Vch', 'Vcv'< / span > , < span class = "org-underline" > ...< / span >
< span class = "org-string" > 'Vnh', 'Vnv'< / span > , < span class = "org-underline" > ...< / span >
< span class = "org-string" > 'Va'< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
uv = load< span class = "org-rainbow-delimiters-depth-1" > (< / span > < span class = "org-string" > 'mat/data_uh.mat'< / span > , < span class = "org-underline" > ...< / span >
< span class = "org-string" > 't', 'Uch', 'Ucv'< / span > , < span class = "org-underline" > ...< / span >
< span class = "org-string" > 'Unh', 'Unv'< / span > , < span class = "org-underline" > ...< / span >
< span class = "org-string" > 'Vph', 'Vpv'< / span > , < span class = "org-underline" > ...< / span >
< span class = "org-string" > 'Vch', 'Vcv'< / span > , < span class = "org-underline" > ...< / span >
< span class = "org-string" > 'Vnh', 'Vnv'< / span > , < span class = "org-underline" > ...< / span >
< span class = "org-string" > 'Va'< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
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< / pre >
< / div >
< p >
We remove the first seconds where the Cercalo is turned on.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > t0 = < span class = "org-highlight-numbers-number" > 1< / span > ;
uh.Uch< span class = "org-rainbow-delimiters-depth-1" > (< / span > t< span class = "org-type" > < < / span > t0< span class = "org-rainbow-delimiters-depth-1" > )< / span > = < span class = "org-rainbow-delimiters-depth-1" > []< / span > ;
uh.Ucv< span class = "org-rainbow-delimiters-depth-1" > (< / span > t< span class = "org-type" > < < / span > t0< span class = "org-rainbow-delimiters-depth-1" > )< / span > = < span class = "org-rainbow-delimiters-depth-1" > []< / span > ;
uh.Unh< span class = "org-rainbow-delimiters-depth-1" > (< / span > t< span class = "org-type" > < < / span > t0< span class = "org-rainbow-delimiters-depth-1" > )< / span > = < span class = "org-rainbow-delimiters-depth-1" > []< / span > ;
uh.Unv< span class = "org-rainbow-delimiters-depth-1" > (< / span > t< span class = "org-type" > < < / span > t0< span class = "org-rainbow-delimiters-depth-1" > )< / span > = < span class = "org-rainbow-delimiters-depth-1" > []< / span > ;
uh.Vph< span class = "org-rainbow-delimiters-depth-1" > (< / span > t< span class = "org-type" > < < / span > t0< span class = "org-rainbow-delimiters-depth-1" > )< / span > = < span class = "org-rainbow-delimiters-depth-1" > []< / span > ;
uh.Vpv< span class = "org-rainbow-delimiters-depth-1" > (< / span > t< span class = "org-type" > < < / span > t0< span class = "org-rainbow-delimiters-depth-1" > )< / span > = < span class = "org-rainbow-delimiters-depth-1" > []< / span > ;
uh.Vch< span class = "org-rainbow-delimiters-depth-1" > (< / span > t< span class = "org-type" > < < / span > t0< span class = "org-rainbow-delimiters-depth-1" > )< / span > = < span class = "org-rainbow-delimiters-depth-1" > []< / span > ;
uh.Vcv< span class = "org-rainbow-delimiters-depth-1" > (< / span > t< span class = "org-type" > < < / span > t0< span class = "org-rainbow-delimiters-depth-1" > )< / span > = < span class = "org-rainbow-delimiters-depth-1" > []< / span > ;
uh.Vnh< span class = "org-rainbow-delimiters-depth-1" > (< / span > t< span class = "org-type" > < < / span > t0< span class = "org-rainbow-delimiters-depth-1" > )< / span > = < span class = "org-rainbow-delimiters-depth-1" > []< / span > ;
uh.Vnv< span class = "org-rainbow-delimiters-depth-1" > (< / span > t< span class = "org-type" > < < / span > t0< span class = "org-rainbow-delimiters-depth-1" > )< / span > = < span class = "org-rainbow-delimiters-depth-1" > []< / span > ;
uh.Va< span class = "org-rainbow-delimiters-depth-1" > (< / span > t< span class = "org-type" > < < / span > t0< span class = "org-rainbow-delimiters-depth-1" > )< / span > = < span class = "org-rainbow-delimiters-depth-1" > []< / span > ;
uh.t< span class = "org-rainbow-delimiters-depth-1" > (< / span > t< span class = "org-type" > < < / span > t0< span class = "org-rainbow-delimiters-depth-1" > )< / span > = < span class = "org-rainbow-delimiters-depth-1" > []< / span > ;
uh.t = uh.t < span class = "org-type" > -< / span > uh.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 > ; % We start at t=< span class = "org-highlight-numbers-number" > 0< / span >
t0 = < span class = "org-highlight-numbers-number" > 1< / span > ;
uv.Uch< span class = "org-rainbow-delimiters-depth-1" > (< / span > t< span class = "org-type" > < < / span > t0< span class = "org-rainbow-delimiters-depth-1" > )< / span > = < span class = "org-rainbow-delimiters-depth-1" > []< / span > ;
uv.Ucv< span class = "org-rainbow-delimiters-depth-1" > (< / span > t< span class = "org-type" > < < / span > t0< span class = "org-rainbow-delimiters-depth-1" > )< / span > = < span class = "org-rainbow-delimiters-depth-1" > []< / span > ;
uv.Unh< span class = "org-rainbow-delimiters-depth-1" > (< / span > t< span class = "org-type" > < < / span > t0< span class = "org-rainbow-delimiters-depth-1" > )< / span > = < span class = "org-rainbow-delimiters-depth-1" > []< / span > ;
uv.Unv< span class = "org-rainbow-delimiters-depth-1" > (< / span > t< span class = "org-type" > < < / span > t0< span class = "org-rainbow-delimiters-depth-1" > )< / span > = < span class = "org-rainbow-delimiters-depth-1" > []< / span > ;
uv.Vph< span class = "org-rainbow-delimiters-depth-1" > (< / span > t< span class = "org-type" > < < / span > t0< span class = "org-rainbow-delimiters-depth-1" > )< / span > = < span class = "org-rainbow-delimiters-depth-1" > []< / span > ;
uv.Vpv< span class = "org-rainbow-delimiters-depth-1" > (< / span > t< span class = "org-type" > < < / span > t0< span class = "org-rainbow-delimiters-depth-1" > )< / span > = < span class = "org-rainbow-delimiters-depth-1" > []< / span > ;
uv.Vch< span class = "org-rainbow-delimiters-depth-1" > (< / span > t< span class = "org-type" > < < / span > t0< span class = "org-rainbow-delimiters-depth-1" > )< / span > = < span class = "org-rainbow-delimiters-depth-1" > []< / span > ;
uv.Vcv< span class = "org-rainbow-delimiters-depth-1" > (< / span > t< span class = "org-type" > < < / span > t0< span class = "org-rainbow-delimiters-depth-1" > )< / span > = < span class = "org-rainbow-delimiters-depth-1" > []< / span > ;
uv.Vnh< span class = "org-rainbow-delimiters-depth-1" > (< / span > t< span class = "org-type" > < < / span > t0< span class = "org-rainbow-delimiters-depth-1" > )< / span > = < span class = "org-rainbow-delimiters-depth-1" > []< / span > ;
uv.Vnv< span class = "org-rainbow-delimiters-depth-1" > (< / span > t< span class = "org-type" > < < / span > t0< span class = "org-rainbow-delimiters-depth-1" > )< / span > = < span class = "org-rainbow-delimiters-depth-1" > []< / span > ;
uv.Va< span class = "org-rainbow-delimiters-depth-1" > (< / span > t< span class = "org-type" > < < / span > t0< span class = "org-rainbow-delimiters-depth-1" > )< / span > = < span class = "org-rainbow-delimiters-depth-1" > []< / span > ;
uv.t< span class = "org-rainbow-delimiters-depth-1" > (< / span > t< span class = "org-type" > < < / span > t0< span class = "org-rainbow-delimiters-depth-1" > )< / span > = < span class = "org-rainbow-delimiters-depth-1" > []< / span > ;
uv.t = uv.t < span class = "org-type" > -< / span > uv.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 > ; % We start at t=< span class = "org-highlight-numbers-number" > 0< / span >
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< / pre >
< / div >
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< div id = "orga6bc5e6" class = "figure" >
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< p > < img src = "figs/identification_uh.png" alt = "identification_uh.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 10: < / span > Identification signals when exciting the horizontal direction (< a href = "./figs/identification_uh.png" > png< / a > , < a href = "./figs/identification_uh.pdf" > pdf< / a > )< / p >
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< / div >
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< div id = "org82bc95e" class = "figure" >
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< p > < img src = "figs/identification_uv.png" alt = "identification_uv.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 11: < / span > Identification signals when exciting in the vertical direction (< a href = "./figs/identification_uv.png" > png< / a > , < a href = "./figs/identification_uv.pdf" > pdf< / a > )< / p >
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< / div >
< / div >
< / div >
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< div id = "outline-container-org0556947" class = "outline-3" >
< h3 id = "org0556947" > < span class = "section-number-3" > 2.5< / span > Estimation of the Frequency Response Function Matrix< / h3 >
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< div class = "outline-text-3" id = "text-2-5" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > win = hanning< span class = "org-rainbow-delimiters-depth-1" > (< / span > ceil< span class = "org-rainbow-delimiters-depth-2" > (< / span > < span class = "org-highlight-numbers-number" > 1< / span > < span class = "org-type" > *< / span > fs< span class = "org-rainbow-delimiters-depth-2" > )< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
< / pre >
< / div >
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< p >
We compute an estimate of the transfer functions.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-rainbow-delimiters-depth-1" > [< / span > tf_Uch_Vph, f< span class = "org-rainbow-delimiters-depth-1" > ]< / span > = tfestimate< span class = "org-rainbow-delimiters-depth-1" > (< / span > uh.Uch, uh.Vph, win, < 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_Uch_Vpv, < span class = "org-type" > ~< / span > < span class = "org-rainbow-delimiters-depth-1" > ]< / span > = tfestimate< span class = "org-rainbow-delimiters-depth-1" > (< / span > uh.Uch, uh.Vpv, win, < 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_Ucv_Vph, < span class = "org-type" > ~< / span > < span class = "org-rainbow-delimiters-depth-1" > ]< / span > = tfestimate< span class = "org-rainbow-delimiters-depth-1" > (< / span > uv.Ucv, uv.Vph, win, < 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_Ucv_Vpv, < span class = "org-type" > ~< / span > < span class = "org-rainbow-delimiters-depth-1" > ]< / span > = tfestimate< span class = "org-rainbow-delimiters-depth-1" > (< / span > uv.Ucv, uv.Vpv, win, < 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 > ;
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< / pre >
< / div >
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< div id = "org171e666" class = "figure" >
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< p > < img src = "figs/frequency_response_matrix.png" alt = "frequency_response_matrix.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 12: < / span > Frequency Response Matrix (< a href = "./figs/frequency_response_matrix.png" > png< / a > , < a href = "./figs/frequency_response_matrix.pdf" > pdf< / a > )< / p >
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< / div >
< / div >
< / div >
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< div id = "outline-container-org6c025b6" class = "outline-3" >
< h3 id = "org6c025b6" > < span class = "section-number-3" > 2.6< / span > Coherence< / h3 >
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< div class = "outline-text-3" id = "text-2-6" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-rainbow-delimiters-depth-1" > [< / span > coh_Uch_Vph, f< span class = "org-rainbow-delimiters-depth-1" > ]< / span > = mscohere< span class = "org-rainbow-delimiters-depth-1" > (< / span > uh.Uch, uh.Vph, win, < 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_Uch_Vpv, < span class = "org-type" > ~< / span > < span class = "org-rainbow-delimiters-depth-1" > ]< / span > = mscohere< span class = "org-rainbow-delimiters-depth-1" > (< / span > uh.Uch, uh.Vpv, win, < 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_Ucv_Vph, < span class = "org-type" > ~< / span > < span class = "org-rainbow-delimiters-depth-1" > ]< / span > = mscohere< span class = "org-rainbow-delimiters-depth-1" > (< / span > uv.Ucv, uv.Vph, win, < 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_Ucv_Vpv, < span class = "org-type" > ~< / span > < span class = "org-rainbow-delimiters-depth-1" > ]< / span > = mscohere< span class = "org-rainbow-delimiters-depth-1" > (< / span > uv.Ucv, uv.Vpv, win, < 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 > ;
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< / pre >
< / div >
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< div id = "orgd207281" class = "figure" >
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< p > < img src = "figs/identification_coherence.png" alt = "identification_coherence.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 13: < / span > Coherence (< a href = "./figs/identification_coherence.png" > png< / a > , < a href = "./figs/identification_coherence.pdf" > pdf< / a > )< / p >
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< / div >
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< / div >
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< div id = "outline-container-org74eeb16" class = "outline-3" >
< h3 id = "org74eeb16" > < span class = "section-number-3" > 2.7< / span > Extraction of a transfer function matrix< / h3 >
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< div class = "outline-text-3" id = "text-2-7" >
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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" >
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< pre class = "src src-matlab" > freqs_res_uh = < span class = "org-rainbow-delimiters-depth-1" > [< / span > < span class = "org-highlight-numbers-number" > 410< / span > < span class = "org-rainbow-delimiters-depth-1" > ]< / span > ; < span class = "org-comment" > % [Hz]< / span >
freqs_res_uv = < span class = "org-rainbow-delimiters-depth-1" > [< / span > < span class = "org-highlight-numbers-number" > 250< / span > < span class = "org-rainbow-delimiters-depth-1" > ]< / span > ; < span class = "org-comment" > % [Hz]< / span >
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< / 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" >
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< pre class = "src src-matlab" > N = < span class = "org-highlight-numbers-number" > 2< / span > ;
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< / 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 >
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poles_uh = < 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_uh< 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_uh< 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 > ;
poles_uv = < 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_uv< 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_uv< 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 > ;
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< / 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" >
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< pre class = "src src-matlab" > weight_Uch_Vph = coh_Uch_Vph';
weight_Uch_Vpv = coh_Uch_Vpv';
weight_Ucv_Vph = coh_Ucv_Vph';
weight_Ucv_Vpv = coh_Ucv_Vpv';
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< / span > < span class = "org-constant" > < span class = "org-rainbow-delimiters-depth-1" > (< / span > < / span > < span class = "org-constant" > freqs_res< / span > < span class = "org-constant" > < span class = "org-rainbow-delimiters-depth-1" > )< / span > < / span >
weight_Uch_Vph< span class = "org-rainbow-delimiters-depth-1" > (< / span > f< span class = "org-type" > > < / span > < span class = "org-rainbow-delimiters-depth-2" > (< / span > < span class = "org-highlight-numbers-number" > 1< / span > < span class = "org-type" > -< / span > alpha< span class = "org-rainbow-delimiters-depth-2" > )< / span > < span class = "org-type" > *< / span > freqs_res_uh< span class = "org-rainbow-delimiters-depth-2" > (< / span > freq_i< span class = "org-rainbow-delimiters-depth-2" > )< / span > < span class = "org-type" > & < / span > f< span class = "org-type" > < < / span > < span class = "org-rainbow-delimiters-depth-2" > (< / span > < span class = "org-highlight-numbers-number" > 1< / span > < span class = "org-type" > +< / span > alpha< span class = "org-rainbow-delimiters-depth-2" > )< / span > < span class = "org-type" > *< / span > freqs_res_uh< span class = "org-rainbow-delimiters-depth-2" > (< / span > freq_i< span class = "org-rainbow-delimiters-depth-2" > )< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > = < span class = "org-highlight-numbers-number" > 10< / span > ;
weight_Uch_Vpv< span class = "org-rainbow-delimiters-depth-1" > (< / span > f< span class = "org-type" > > < / span > < span class = "org-rainbow-delimiters-depth-2" > (< / span > < span class = "org-highlight-numbers-number" > 1< / span > < span class = "org-type" > -< / span > alpha< span class = "org-rainbow-delimiters-depth-2" > )< / span > < span class = "org-type" > *< / span > freqs_res_uh< span class = "org-rainbow-delimiters-depth-2" > (< / span > freq_i< span class = "org-rainbow-delimiters-depth-2" > )< / span > < span class = "org-type" > & < / span > f< span class = "org-type" > < < / span > < span class = "org-rainbow-delimiters-depth-2" > (< / span > < span class = "org-highlight-numbers-number" > 1< / span > < span class = "org-type" > +< / span > alpha< span class = "org-rainbow-delimiters-depth-2" > )< / span > < span class = "org-type" > *< / span > freqs_res_uh< span class = "org-rainbow-delimiters-depth-2" > (< / span > freq_i< span class = "org-rainbow-delimiters-depth-2" > )< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > = < span class = "org-highlight-numbers-number" > 10< / span > ;
weight_Ucv_Vph< span class = "org-rainbow-delimiters-depth-1" > (< / span > f< span class = "org-type" > > < / span > < span class = "org-rainbow-delimiters-depth-2" > (< / span > < span class = "org-highlight-numbers-number" > 1< / span > < span class = "org-type" > -< / span > alpha< span class = "org-rainbow-delimiters-depth-2" > )< / span > < span class = "org-type" > *< / span > freqs_res_uv< span class = "org-rainbow-delimiters-depth-2" > (< / span > freq_i< span class = "org-rainbow-delimiters-depth-2" > )< / span > < span class = "org-type" > & < / span > f< span class = "org-type" > < < / span > < span class = "org-rainbow-delimiters-depth-2" > (< / span > < span class = "org-highlight-numbers-number" > 1< / span > < span class = "org-type" > +< / span > alpha< span class = "org-rainbow-delimiters-depth-2" > )< / span > < span class = "org-type" > *< / span > freqs_res_uv< span class = "org-rainbow-delimiters-depth-2" > (< / span > freq_i< span class = "org-rainbow-delimiters-depth-2" > )< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > = < span class = "org-highlight-numbers-number" > 10< / span > ;
weight_Ucv_Vpv< span class = "org-rainbow-delimiters-depth-1" > (< / span > f< span class = "org-type" > > < / span > < span class = "org-rainbow-delimiters-depth-2" > (< / span > < span class = "org-highlight-numbers-number" > 1< / span > < span class = "org-type" > -< / span > alpha< span class = "org-rainbow-delimiters-depth-2" > )< / span > < span class = "org-type" > *< / span > freqs_res_uv< span class = "org-rainbow-delimiters-depth-2" > (< / span > freq_i< span class = "org-rainbow-delimiters-depth-2" > )< / span > < span class = "org-type" > & < / span > f< span class = "org-type" > < < / span > < span class = "org-rainbow-delimiters-depth-2" > (< / span > < span class = "org-highlight-numbers-number" > 1< / span > < span class = "org-type" > +< / span > alpha< span class = "org-rainbow-delimiters-depth-2" > )< / span > < span class = "org-type" > *< / span > freqs_res_uv< span class = "org-rainbow-delimiters-depth-2" > (< / span > freq_i< span class = "org-rainbow-delimiters-depth-2" > )< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > = < span class = "org-highlight-numbers-number" > 10< / span > ;
< span class = "org-keyword" > end< / span >
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< / pre >
< / div >
< p >
Ignore data above some frequency.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > weight_Uch_Vph< 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 > ;
weight_Uch_Vpv< 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 > ;
weight_Ucv_Vph< 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 > ;
weight_Ucv_Vpv< 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 > ;
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< / pre >
< / div >
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< div id = "org9848359" class = "figure" >
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< p > < img src = "figs/weights.png" alt = "weights.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 14: < / span > Weights amplitude (< a href = "./figs/weights.png" > png< / a > , < a href = "./figs/weights.pdf" > pdf< / a > )< / p >
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< / 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 >
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< span class = "org-rainbow-delimiters-depth-1" > [< / span > SER_Uch_Vph, poles, < span class = "org-type" > ~< / span > , fit_Uch_Vph< span class = "org-rainbow-delimiters-depth-1" > ]< / span > = vectfit3< span class = "org-rainbow-delimiters-depth-1" > (< / span > tf_Uch_Vph< 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_uh, weight_Uch_Vph, opts< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
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< 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 >
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< span class = "org-rainbow-delimiters-depth-1" > [< / span > SER_Uch_Vpv, poles, < span class = "org-type" > ~< / span > , fit_Uch_Vpv< span class = "org-rainbow-delimiters-depth-1" > ]< / span > = vectfit3< span class = "org-rainbow-delimiters-depth-1" > (< / span > tf_Uch_Vpv< 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_uh, weight_Uch_Vpv, opts< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
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< 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 >
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< span class = "org-rainbow-delimiters-depth-1" > [< / span > SER_Ucv_Vph, poles, < span class = "org-type" > ~< / span > , fit_Ucv_Vph< span class = "org-rainbow-delimiters-depth-1" > ]< / span > = vectfit3< span class = "org-rainbow-delimiters-depth-1" > (< / span > tf_Ucv_Vph< 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_uv, weight_Ucv_Vph, opts< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
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< 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 >
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< span class = "org-rainbow-delimiters-depth-1" > [< / span > SER_Ucv_Vpv, poles, < span class = "org-type" > ~< / span > , fit_Ucv_Vpv< span class = "org-rainbow-delimiters-depth-1" > ]< / span > = vectfit3< span class = "org-rainbow-delimiters-depth-1" > (< / span > tf_Ucv_Vpv< 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_uv, weight_Ucv_Vpv, opts< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
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< span class = "org-keyword" > end< / span >
< / pre >
< / div >
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< div id = "org61a8901" class = "figure" >
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< p > < img src = "figs/identification_matrix_fit.png" alt = "identification_matrix_fit.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 15: < / 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 >
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< / div >
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< div id = "org05ccf64" class = "figure" >
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< p > < img src = "figs/identification_matrix_fit_phase.png" alt = "identification_matrix_fit_phase.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 16: < / span > Transfer Function Extraction of the FRF matrix (< a href = "./figs/identification_matrix_fit_phase.png" > png< / a > , < a href = "./figs/identification_matrix_fit_phase.pdf" > pdf< / a > )< / p >
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< / div >
< p >
And finally, we create the identified state space model:
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > G_uh_xh = 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_uh_xh.A< span class = "org-rainbow-delimiters-depth-3" > )< / span > ,SER_uh_xh.B,SER_uh_xh.C,SER_uh_xh.D< span class = "org-rainbow-delimiters-depth-2" > )< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
G_uv_xh = 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_uv_xh.A< span class = "org-rainbow-delimiters-depth-3" > )< / span > ,SER_uv_xh.B,SER_uv_xh.C,SER_uv_xh.D< span class = "org-rainbow-delimiters-depth-2" > )< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
G_uh_xv = 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_uh_xv.A< span class = "org-rainbow-delimiters-depth-3" > )< / span > ,SER_uh_xv.B,SER_uh_xv.C,SER_uh_xv.D< span class = "org-rainbow-delimiters-depth-2" > )< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
G_uv_xv = 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_uv_xv.A< span class = "org-rainbow-delimiters-depth-3" > )< / span > ,SER_uv_xv.B,SER_uv_xv.C,SER_uv_xv.D< span class = "org-rainbow-delimiters-depth-2" > )< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
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G = < span class = "org-rainbow-delimiters-depth-1" > [< / span > G_uh_xh, G_uv_xh;
G_uh_xv, G_uv_xv< span class = "org-rainbow-delimiters-depth-1" > ]< / span > ;
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< / 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-orge157b63" class = "outline-2" >
< h2 id = "orge157b63" > < span class = "section-number-2" > 3< / span > Calibration of the 4 Quadrant Diode< / h2 >
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< / div >
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< div id = "outline-container-orgc276bb8" class = "outline-2" >
< h2 id = "orgc276bb8" > < span class = "section-number-2" > 4< / span > Plant Scaling< / h2 >
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< div class = "outline-text-2" id = "text-4" >
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< ul class = "org-ul" >
< li > measured noise< / li >
< li > expected perturbations< / li >
< li > maximum input usage< / li >
< li > maximum wanted error< / li >
< / ul >
< / div >
< / div >
< div id = "outline-container-orgffff224" class = "outline-2" >
< h2 id = "orgffff224" > < span class = "section-number-2" > 5< / span > Plant Analysis< / h2 >
< div class = "outline-text-2" id = "text-5" >
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< / div >
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< div id = "outline-container-orge9e5454" class = "outline-3" >
< h3 id = "orge9e5454" > < span class = "section-number-3" > 5.1< / span > Load Plant< / h3 >
< div class = "outline-text-3" id = "text-5-1" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > load< 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 >
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< div id = "outline-container-org425e4e2" class = "outline-3" >
< h3 id = "org425e4e2" > < span class = "section-number-3" > 5.2< / span > RGA-Number< / h3 >
< div class = "outline-text-3" id = "text-5-2" >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > freqs = logspace< span class = "org-rainbow-delimiters-depth-1" > (< / span > < span class = "org-highlight-numbers-number" > 2< / span > , < span class = "org-highlight-numbers-number" > 4< / span > , < span class = "org-highlight-numbers-number" > 1000< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
G_resp = freqresp< span class = "org-rainbow-delimiters-depth-1" > (< / span > G, freqs, < span class = "org-string" > 'Hz'< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
A = zeros< span class = "org-rainbow-delimiters-depth-1" > (< / span > size< span class = "org-rainbow-delimiters-depth-2" > (< / span > G_resp< span class = "org-rainbow-delimiters-depth-2" > )< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
RGAnum = zeros< 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 > freqs< span class = "org-rainbow-delimiters-depth-2" > )< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > i< / span > = < span class = "org-constant" > < span class = "org-highlight-numbers-number" > 1< / span > < / span > < span class = "org-constant" > :length< / span > < span class = "org-constant" > < span class = "org-rainbow-delimiters-depth-1" > (< / span > < / span > < span class = "org-constant" > freqs< / span > < span class = "org-constant" > < span class = "org-rainbow-delimiters-depth-1" > )< / span > < / span >
A< span class = "org-rainbow-delimiters-depth-1" > (< / span > < span class = "org-type" > :< / span > , < span class = "org-type" > :< / span > , < span class = "org-constant" > i< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > = G_resp< span class = "org-rainbow-delimiters-depth-1" > (< / span > < span class = "org-type" > :< / span > , < span class = "org-type" > :< / span > , < span class = "org-constant" > i< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > < span class = "org-type" > .*< / span > inv< span class = "org-rainbow-delimiters-depth-1" > (< / span > G_resp< span class = "org-rainbow-delimiters-depth-2" > (< / span > < span class = "org-type" > :< / span > , < span class = "org-type" > :< / span > , < span class = "org-constant" > i< / span > < span class = "org-rainbow-delimiters-depth-2" > )< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > ';
RGAnum< span class = "org-rainbow-delimiters-depth-1" > (< / span > < span class = "org-constant" > i< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > = sum< span class = "org-rainbow-delimiters-depth-1" > (< / span > sum< span class = "org-rainbow-delimiters-depth-2" > (< / span > abs< span class = "org-rainbow-delimiters-depth-3" > (< / span > A< span class = "org-rainbow-delimiters-depth-4" > (< / span > < span class = "org-type" > :< / span > , < span class = "org-type" > :< / span > , < span class = "org-constant" > i< / span > < span class = "org-rainbow-delimiters-depth-4" > )< / span > < span class = "org-type" > -< / span > eye< span class = "org-rainbow-delimiters-depth-4" > (< / span > < span class = "org-highlight-numbers-number" > 2< / span > < span class = "org-rainbow-delimiters-depth-4" > )< / span > < span class = "org-rainbow-delimiters-depth-3" > )< / span > < span class = "org-rainbow-delimiters-depth-2" > )< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
< span class = "org-keyword" > end< / span >
< span class = "org-comment" > % < / span > < span class = "org-comment" > < span class = "org-constant" > RGA < / span > < / span > < span class = "org-comment" > = G0.*inv(G0)';< / span >
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-type" > figure< / span > ;
plot< span class = "org-rainbow-delimiters-depth-1" > (< / span > freqs, RGAnum< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
< 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-rainbow-delimiters-depth-1" > )< / span > ;
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > U = zeros< span class = "org-rainbow-delimiters-depth-1" > (< / span > < span class = "org-highlight-numbers-number" > 2< / span > , < span class = "org-highlight-numbers-number" > 2< / span > , length< span class = "org-rainbow-delimiters-depth-2" > (< / span > freqs< span class = "org-rainbow-delimiters-depth-2" > )< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
S = zeros< span class = "org-rainbow-delimiters-depth-1" > (< / span > < span class = "org-highlight-numbers-number" > 2< / span > , < span class = "org-highlight-numbers-number" > 2< / span > , length< span class = "org-rainbow-delimiters-depth-2" > (< / span > freqs< span class = "org-rainbow-delimiters-depth-2" > )< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span >
V = zeros< span class = "org-rainbow-delimiters-depth-1" > (< / span > < span class = "org-highlight-numbers-number" > 2< / span > , < span class = "org-highlight-numbers-number" > 2< / span > , length< span class = "org-rainbow-delimiters-depth-2" > (< / span > freqs< span class = "org-rainbow-delimiters-depth-2" > )< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > i< / span > = < span class = "org-constant" > < span class = "org-highlight-numbers-number" > 1< / span > < / span > < span class = "org-constant" > :length< / span > < span class = "org-constant" > < span class = "org-rainbow-delimiters-depth-1" > (< / span > < / span > < span class = "org-constant" > freqs< / span > < span class = "org-constant" > < span class = "org-rainbow-delimiters-depth-1" > )< / span > < / span >
< span class = "org-rainbow-delimiters-depth-1" > [< / span > Ui, Si, Vi< span class = "org-rainbow-delimiters-depth-1" > ]< / span > = svd< span class = "org-rainbow-delimiters-depth-1" > (< / span > G_resp< span class = "org-rainbow-delimiters-depth-2" > (< / span > < span class = "org-type" > :< / span > , < span class = "org-type" > :< / span > , < span class = "org-constant" > i< / span > < span class = "org-rainbow-delimiters-depth-2" > )< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
U< span class = "org-rainbow-delimiters-depth-1" > (< / span > < span class = "org-type" > :< / span > , < span class = "org-type" > :< / span > , < span class = "org-constant" > i< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > = Ui;
S< span class = "org-rainbow-delimiters-depth-1" > (< / span > < span class = "org-type" > :< / span > , < span class = "org-type" > :< / span > , < span class = "org-constant" > i< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > = Si;
V< span class = "org-rainbow-delimiters-depth-1" > (< / span > < span class = "org-type" > :< / span > , < span class = "org-type" > :< / span > , < span class = "org-constant" > i< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > = Vi;
< span class = "org-keyword" > end< / span >
< / pre >
< / div >
< / div >
< / div >
2019-09-16 18:14:44 +02:00
< div id = "outline-container-orge80bcc0" class = "outline-3" >
< h3 id = "orge80bcc0" > < span class = "section-number-3" > 5.3< / span > Rotation Matrix< / h3 >
< div class = "outline-text-3" id = "text-5-3" >
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< div class = "org-src-container" >
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< 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 >
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< div id = "outline-container-orgfd824b0" class = "outline-2" >
< h2 id = "orgfd824b0" > < span class = "section-number-2" > 6< / span > Control Objective< / h2 >
< div class = "outline-text-2" id = "text-6" >
2019-09-12 15:50:31 +02:00
< 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 >
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< div id = "outline-container-orgc7afa66" class = "outline-2" >
< h2 id = "orgc7afa66" > < span class = "section-number-2" > 7< / span > Control Design< / h2 >
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< / div >
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< div id = "outline-container-org5791267" class = "outline-2" >
< h2 id = "org5791267" > < span class = "section-number-2" > 8< / span > Measurement of the non-repeatability< / h2 >
2019-09-10 18:16:16 +02:00
< / div >
< / div >
< div id = "postamble" class = "status" >
< p class = "author" > Author: Dehaeze Thomas< / p >
2019-09-16 18:14:44 +02:00
< p class = "date" > Created: 2019-09-16 lun. 15:21< / 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 >
