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< / div > < div id = "content" >
< h1 class = "title" > Control Requirements< / 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 = "#org0341df1" > 1. Simplify Model for the Nano-Hexapod< / a >
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< ul >
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< li > < a href = "#org136c9af" > 1.1. Model of the nano-hexapod< / a > < / li >
< li > < a href = "#org2fbecfd" > 1.2. How to include Ground Motion in the model?< / a > < / li >
< li > < a href = "#org8c1e462" > 1.3. Motion of the micro-station< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org92b1239" > 2. Control with the Stiff Nano-Hexapod< / a >
< ul >
< li > < a href = "#org19b83b7" > 2.1. Definition of the values< / a > < / li >
< li > < a href = "#org0e9811a" > 2.2. Control using \(d\)< / a >
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< ul >
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< li > < a href = "#org9828aed" > 2.2.1. Control Architecture< / a > < / li >
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< li > < a href = "#org5a120e1" > 2.2.2. Analytical Analysis< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orga741e48" > 2.3. Control using \(F_m\)< / a >
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< ul >
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< li > < a href = "#org4c74e08" > 2.3.1. Control Architecture< / a > < / li >
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< li > < a href = "#orgdd5134e" > 2.3.2. Pure Integrator< / a > < / li >
< li > < a href = "#org5011ab0" > 2.3.3. Low pass filter< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org4fce174" > 2.4. Comparison< / a > < / li >
< li > < a href = "#org5e0585d" > 2.5. Control using \(x\)< / a >
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< ul >
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< li > < a href = "#orgfab8395" > 2.5.1. Analytical analysis< / a > < / li >
< li > < a href = "#org625e3c2" > 2.5.2. Control implementation< / a > < / li >
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< li > < a href = "#org3db77f5" > 2.5.3. Results< / a > < / li >
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< / ul >
< / li >
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< / ul >
< / li >
< li > < a href = "#org7c4b4fc" > 3. Comparison with the use of a Soft nano-hexapod< / a > < / li >
< li > < a href = "#orgc0253c3" > 4. Estimate the level of vibration< / a > < / li >
< li > < a href = "#org764c4a9" > 5. Requirements on the norm of closed-loop transfer functions< / a >
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< ul >
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< li > < a href = "#org27379f3" > 5.1. Approximation of the ASD of perturbations< / a > < / li >
< li > < a href = "#orgff3d823" > 5.2. Wanted ASD of outputs< / a > < / li >
< li > < a href = "#org8c6b37c" > 5.3. Limiting the bandwidth< / a > < / li >
< li > < a href = "#org50054f2" > 5.4. Generalized Weighted plant< / a > < / li >
< li > < a href = "#org949ab66" > 5.5. Synthesis< / a > < / li >
< li > < a href = "#orgfe970e4" > 5.6. Loop Gain< / a > < / li >
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< li > < a href = "#org5211e8c" > 5.7. Results< / a > < / li >
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< li > < a href = "#orgb18d7df" > 5.8. Requirements< / a > < / li >
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< / ul >
< / li >
< / ul >
< / div >
< / div >
< p >
The goal here is to write clear specifications for the NASS.
< / p >
< p >
This can then be used for the control synthesis and for the design of the nano-hexapod.
< / p >
< p >
Ideal, specifications on the norm of closed loop transfer function should be written.
< / p >
< div id = "outline-container-org0341df1" class = "outline-2" >
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< h2 id = "org0341df1" > < span class = "section-number-2" > 1< / span > Simplify Model for the Nano-Hexapod< / h2 >
< div class = "outline-text-2" id = "text-1" >
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< / div >
< div id = "outline-container-org136c9af" class = "outline-3" >
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< h3 id = "org136c9af" > < span class = "section-number-3" > 1.1< / span > Model of the nano-hexapod< / h3 >
< div class = "outline-text-3" id = "text-1-1" >
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< p >
Let’ s consider the simple mechanical system in Figure < a href = "#orgfa3391a" > 1< / a > .
< / p >
< div id = "orgfa3391a" class = "figure" >
< p > < img src = "figs/nass_simple_model.png" alt = "nass_simple_model.png" / >
< / p >
< p > < span class = "figure-number" > Figure 1: < / span > Simplified mechanical system for the nano-hexapod< / p >
< / div >
< p >
The signals are described in table < a href = "#orgd89e830" > 1< / a > .
< / p >
< table id = "orgd89e830" border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
< caption class = "t-above" > < span class = "table-number" > Table 1:< / span > Signals definition for the generalized plant< / caption >
< colgroup >
< col class = "org-left" / >
< col class = "org-left" / >
< col class = "org-left" / >
< / colgroup >
< thead >
< tr >
< th scope = "col" class = "org-left" >   < / th >
< th scope = "col" class = "org-left" > < b > Symbol< / b > < / th >
< th scope = "col" class = "org-left" > < b > Meaning< / b > < / th >
< / tr >
< / thead >
< tbody >
< tr >
< td class = "org-left" > < b > Exogenous Inputs< / b > < / td >
< td class = "org-left" > \(x_\mu\)< / td >
< td class = "org-left" > Motion of the $ν $-hexapod’ s base< / td >
< / tr >
< tr >
< td class = "org-left" >   < / td >
< td class = "org-left" > \(F_d\)< / td >
< td class = "org-left" > External Forces applied to the Payload< / td >
< / tr >
< tr >
< td class = "org-left" >   < / td >
< td class = "org-left" > \(r\)< / td >
< td class = "org-left" > Reference signal for tracking< / td >
< / tr >
< / tbody >
< tbody >
< tr >
< td class = "org-left" > < b > Exogenous Outputs< / b > < / td >
< td class = "org-left" > \(x\)< / td >
< td class = "org-left" > Absolute Motion of the Payload< / td >
< / tr >
< / tbody >
< tbody >
< tr >
< td class = "org-left" > < b > Sensed Outputs< / b > < / td >
< td class = "org-left" > \(F_m\)< / td >
< td class = "org-left" > Force Sensors in each leg< / td >
< / tr >
< tr >
< td class = "org-left" >   < / td >
< td class = "org-left" > \(d\)< / td >
< td class = "org-left" > Measured displacement of each leg< / td >
< / tr >
< tr >
< td class = "org-left" >   < / td >
< td class = "org-left" > \(x\)< / td >
< td class = "org-left" > Absolute Motion of the Payload< / td >
< / tr >
< / tbody >
< tbody >
< tr >
< td class = "org-left" > < b > Control Signals< / b > < / td >
< td class = "org-left" > \(F\)< / td >
< td class = "org-left" > Actuator Inputs< / td >
< / tr >
< / tbody >
< / table >
< p >
For the nano-hexapod alone, we have the following equations:
\[ \begin{align*}
x & = \frac{1}{ms^2 + k} F + \frac{1}{ms^2 + k} F_d + \frac{k}{ms^2 + k} x_\mu \\
F_m & = \frac{ms^2}{ms^2 + k} F - \frac{k}{ms^2 + k} F_d + \frac{k m s^2}{ms^2 + k} x_\mu \\
d & = \frac{1}{ms^2 + k} F + \frac{1}{ms^2 + k} F_d - \frac{ms^2}{ms^2 + k} x_\mu
\end{align*} \]
< / p >
< p >
We can write the equations function of \(\omega_\nu = \sqrt{\frac{k}{m}}\):
\[ \begin{align*}
x & = \frac{1/k}{1 + \frac{s^2}{\omega_\nu^2}} F + \frac{1/k}{1 + \frac{s^2}{\omega_\nu^2}} F_d + \frac{1}{1 + \frac{s^2}{\omega_\nu^2}} x_\mu \\
F_m & = \frac{\frac{s^2}{\omega_\nu^2}}{1 + \frac{s^2}{\omega_\nu^2}} F - \frac{1}{1 + \frac{s^2}{\omega_\nu^2}} F_d + \frac{k \frac{s^2}{\omega_\nu^2}}{1 + \frac{s^2}{\omega_\nu^2}} x_\mu \\
d & = \frac{1/k}{1 + \frac{s^2}{\omega_\nu^2}} F + \frac{1/k}{1 + \frac{s^2}{\omega_\nu^2}} F_d - \frac{\frac{s^2}{\omega_\nu^2}}{1 + \frac{s^2}{\omega_\nu^2}} x_\mu
\end{align*} \]
< / p >
< p >
< b > Assumptions< / b > :
< / p >
< ul class = "org-ul" >
< li > the forces applied by the nano-hexapod have no influence on the micro-station, specifically on the displacement of the top platform of the micro-hexapod.< / li >
< / ul >
< p >
This means that the nano-hexapod can be considered separately from the micro-station and that the motion \(x_\mu\) is imposed and considered as an external input.
< / p >
< p >
The nano-hexapod can thus be represented as in Figure < a href = "#orgb2d1168" > 2< / a > .
< / p >
< div id = "orgb2d1168" class = "figure" >
< p > < img src = "figs/nano_station_inputs_outputs.png" alt = "nano_station_inputs_outputs.png" / >
< / p >
< p > < span class = "figure-number" > Figure 2: < / span > Block representation of the nano-hexapod< / p >
< / div >
< / div >
< / div >
< div id = "outline-container-org2fbecfd" class = "outline-3" >
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< h3 id = "org2fbecfd" > < span class = "section-number-3" > 1.2< / span > How to include Ground Motion in the model?< / h3 >
< div class = "outline-text-3" id = "text-1-2" >
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< p >
What we measure is not the absolute motion \(x\), but the relative motion \(x - w\) where \(w\) is the motion of the granite.
< / p >
< p >
Also, \(w\) induces some motion \(x_\mu\) through the transmissibility of the micro-station.
< / p >
< / div >
< / div >
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< div id = "outline-container-org8c1e462" class = "outline-3" >
< h3 id = "org8c1e462" > < span class = "section-number-3" > 1.3< / span > Motion of the micro-station< / h3 >
< div class = "outline-text-3" id = "text-1-3" >
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< p >
As explained, we consider \(x_\mu\) as an external input (\(F\) has no influence on \(x_\mu\)).
< / p >
< p >
\(x_\mu\) is the motion of the micro-station’ s top platform due to the motion of each stage of the micro-station.
< / p >
< p >
We consider that \(x_\mu\) has the following form:
\[ x_\mu = T_\mu r + d_\mu \]
where:
< / p >
< ul class = "org-ul" >
< li > \(T_\mu r\) corresponds to the response of the stages due to the reference \(r\)< / li >
< li > \(d_\mu\) is the motion of the hexapod due to all the vibrations of the stages< / li >
< / ul >
< p >
\(T_\mu\) can be considered to be a low pass filter with a bandwidth corresponding approximatively to the bandwidth of the micro-station’ s stages.
To simplify, we can consider \(T_\mu\) to be a first order low pass filter:
\[ T_\mu = \frac{1}{1 + s/\omega_\mu} \]
where \(\omega_\mu\) corresponds to the tracking speed of the micro-station.
< / p >
< p >
What is important to note is that while \(x_\mu\) is viewed as a perturbation from the nano-hexapod point of view, \(x_\mu\) < b > does< / b > depend on the reference signal \(r\).
< / p >
< p >
Also, here, we suppose that the granite is not moving.
< / p >
< p >
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If we now include the motion of the granite \(w\), we obtain the block diagram shown in Figure < a href = "#org974c98f" > 3< / a > .
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< / p >
< div id = "org974c98f" class = "figure" >
< p > < img src = "figs/nano_station_ground_motion.png" alt = "nano_station_ground_motion.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 3: < / span > Ground Motion \(w\) included< / p >
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< / div >
< p >
\(T_w\) is the mechanical transmissibility of the micro-station.
We can approximate this transfer function by a second order low pass filter:
\[ T_w = \frac{1}{1 + 2 \xi s/\omega_0 + s^2/\omega_0^2} \]
< / p >
< / div >
< / div >
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< / div >
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< div id = "outline-container-org92b1239" class = "outline-2" >
< h2 id = "org92b1239" > < span class = "section-number-2" > 2< / span > Control with the Stiff Nano-Hexapod< / h2 >
< div class = "outline-text-2" id = "text-2" >
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< / div >
< div id = "outline-container-org19b83b7" class = "outline-3" >
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< h3 id = "org19b83b7" > < span class = "section-number-3" > 2.1< / span > Definition of the values< / h3 >
< div class = "outline-text-3" id = "text-2-1" >
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< p >
Let’ s define the mass and stiffness of the nano-hexapod.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > m = 50; < span class = "org-comment" > % [kg]< / span >
k = 1e7; < span class = "org-comment" > % [N/m]< / span >
< / pre >
< / div >
< p >
Let’ s define the Plant as shown in Figure < a href = "#orgb2d1168" > 2< / a > :
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Gn = 1< span class = "org-type" > /< / span > (m< span class = "org-type" > *< / span > s< span class = "org-type" > ^< / span > 2 < span class = "org-type" > +< / span > k)< span class = "org-type" > *< / span > [< span class = "org-type" > -< / span > k, k< span class = "org-type" > *< / span > m< span class = "org-type" > *< / span > s< span class = "org-type" > ^< / span > 2, m< span class = "org-type" > *< / span > s< span class = "org-type" > ^< / span > 2; 1, < span class = "org-type" > -< / span > m< span class = "org-type" > *< / span > s< span class = "org-type" > ^< / span > 2, 1; 1, k, 1];
Gn.InputName = {< span class = "org-string" > 'Fd'< / span > , < span class = "org-string" > 'xmu'< / span > , < span class = "org-string" > 'F'< / span > };
Gn.OutputName = {< span class = "org-string" > 'Fm'< / span > , < span class = "org-string" > 'd'< / span > , < span class = "org-string" > 'x'< / span > };
< / pre >
< / div >
< p >
Now, define the transmissibility transfer function \(T_\mu\) corresponding to the micro-station motion.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > wmu = 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > 50; < span class = "org-comment" > % [rad/s]< / span >
Tmu = 1< span class = "org-type" > /< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > wmu);
Tmu.InputName = {< span class = "org-string" > 'r1'< / span > };
Tmu.OutputName = {< span class = "org-string" > 'ymu'< / span > };
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > w0 = 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > 40;
xi = 0.5;
Tw = 1< span class = "org-type" > /< / span > (1 < span class = "org-type" > +< / span > 2< span class = "org-type" > *< / span > xi< span class = "org-type" > *< / span > s< span class = "org-type" > /< / span > w0 < span class = "org-type" > +< / span > s< span class = "org-type" > ^< / span > 2< span class = "org-type" > /< / span > w0< span class = "org-type" > ^< / span > 2);
Tw.InputName = {< span class = "org-string" > 'w1'< / span > };
Tw.OutputName = {< span class = "org-string" > 'dw'< / span > };
< / pre >
< / div >
< p >
We add the fact that \(x_\mu = d_\mu + T_\mu r + T_w w\):
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Wsplit = [tf(1); tf(1)];
Wsplit.InputName = {< span class = "org-string" > 'w'< / span > };
Wsplit.OutputName = {< span class = "org-string" > 'w1'< / span > , < span class = "org-string" > 'w2'< / span > };
S = sumblk(< span class = "org-string" > 'xmu = ymu + dmu + dw'< / span > );
Sw = sumblk(< span class = "org-string" > 'y = x - w2'< / span > );
Gpz = connect(Gn, S, Wsplit, Tw, Tmu, Sw, {< span class = "org-string" > 'Fd'< / span > , < span class = "org-string" > 'dmu'< / span > , < span class = "org-string" > 'r1'< / span > , < span class = "org-string" > 'F'< / span > , < span class = "org-string" > 'w'< / span > }, {< span class = "org-string" > 'Fm'< / span > , < span class = "org-string" > ' d ' < / span > , < span class = "org-string" > 'y'< / span > });
< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org0e9811a" class = "outline-3" >
< h3 id = "org0e9811a" > < span class = "section-number-3" > 2.2< / span > Control using \(d\)< / h3 >
< div class = "outline-text-3" id = "text-2-2" >
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< / div >
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< div id = "outline-container-org9828aed" class = "outline-4" >
< h4 id = "org9828aed" > < span class = "section-number-4" > 2.2.1< / span > Control Architecture< / h4 >
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< div class = "outline-text-4" id = "text-2-2-1" >
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< p >
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Let’ s consider a feedback loop using \(d\) as shown in Figure < a href = "#orgb50386a" > 4< / a > .
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< / p >
< div id = "orgb50386a" class = "figure" >
< p > < img src = "figs/nano_station_control_d.png" alt = "nano_station_control_d.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 4: < / span > Feedback diagram using \(d\)< / p >
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< / div >
< / div >
< / div >
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< div id = "outline-container-org5a120e1" class = "outline-4" >
< h4 id = "org5a120e1" > < span class = "section-number-4" > 2.2.2< / span > Analytical Analysis< / h4 >
< div class = "outline-text-4" id = "text-2-2-2" >
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< p >
Let’ s apply a direct velocity feedback by deriving \(d\):
\[ F = F^\prime - g s d \]
< / p >
< p >
Thus:
\[ d = \frac{1}{ms^2 + gs + k} F^\prime + \frac{1}{ms^2 + gs + k} F_d - \frac{ms^2}{ms^2 + gs + k} x_\mu \]
< / p >
< p >
\[ F = \frac{ms^2 + k}{ms^2 + gs + k} F^\prime - \frac{gs}{ms^2 + gs + k} F_d + \frac{mgs^3}{ms^2 + gs + k} x_\mu \]
< / p >
< p >
and
\[ x = \frac{1}{ms^2 + k} (\frac{ms^2 + k}{ms^2 + gs + k} F^\prime - \frac{gs}{ms^2 + gs + k} F_d + \frac{mgs^3}{ms^2 + gs + k} x_\mu) + \frac{1}{ms^2 + k} F_d + \frac{k}{ms^2 + k} x_\mu \]
< / p >
< p >
\[ x = \frac{ms^2 + k}{(ms^2 + k) (ms^2 + gs + k)} F^\prime + \frac{ms^2 + k}{(ms^2 + k) (ms^2 + gs + k)} F_d + \frac{mgs^3 + k(ms^2 + gs + k)}{(ms^2 + k) (ms^2 + gs + k)} x_\mu \]
< / p >
< p >
And we finally obtain:
\[ x = \frac{1}{ms^2 + gs + k} F^\prime + \frac{1}{ms^2 + gs + k} F_d + \frac{gs + k}{ms^2 + gs + k} x_\mu \]
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > K_dvf = 2< span class = "org-type" > *< / span > sqrt(k< span class = "org-type" > *< / span > m)< span class = "org-type" > *< / span > s;
K_dvf.InputName = {< span class = "org-string" > 'd'< / span > };
K_dvf.OutputName = {< span class = "org-string" > 'F'< / span > };
Gpz_dvf = feedback(Gpz, K_dvf, < span class = "org-string" > 'name'< / span > );
< / pre >
< / div >
< p >
Now let’ s consider that \(x_\mu = d_\mu + T_\mu r\)
< / p >
< p >
\[ x = \frac{1}{ms^2 + gs + k} F^\prime + \frac{1}{ms^2 + gs + k} F_d + \frac{gs + k}{ms^2 + gs + k} d_\mu + T_\mu \frac{gs + k}{ms^2 + gs + k} r \]
< / p >
< p >
And \(\epsilon = r - x\):
\[ \epsilon = \frac{1}{ms^2 + gs + k} F^\prime + \frac{1}{ms^2 + gs + k} F_d + \frac{gs + k}{ms^2 + gs + k} d_\mu + \frac{ms^2 + gs + k - T_\mu (gs + k)}{ms^2 + gs + k} r \]
< / p >
< div class = "important" >
< p >
\[ \epsilon = \frac{1}{ms^2 + gs + k} F^\prime + \frac{1}{ms^2 + gs + k} F_d + \frac{gs + k}{ms^2 + gs + k} d_\mu + \frac{ms^2 - S_\mu(gs + k)}{ms^2 + gs + k} r \]
< / p >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-orga741e48" class = "outline-3" >
< h3 id = "orga741e48" > < span class = "section-number-3" > 2.3< / span > Control using \(F_m\)< / h3 >
< div class = "outline-text-3" id = "text-2-3" >
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< / div >
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< div id = "outline-container-org4c74e08" class = "outline-4" >
< h4 id = "org4c74e08" > < span class = "section-number-4" > 2.3.1< / span > Control Architecture< / h4 >
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< div class = "outline-text-4" id = "text-2-3-1" >
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< p >
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Let’ s consider a feedback loop using \(Fm\) as shown in Figure < a href = "#org5012ef2" > 5< / a > .
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< / p >
< div id = "org5012ef2" class = "figure" >
< p > < img src = "figs/nano_station_control_Fm.png" alt = "nano_station_control_Fm.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 5: < / span > Feedback diagram using \(F_m\)< / p >
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< / div >
< / div >
< / div >
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< div id = "outline-container-orgdd5134e" class = "outline-4" >
< h4 id = "orgdd5134e" > < span class = "section-number-4" > 2.3.2< / span > Pure Integrator< / h4 >
< div class = "outline-text-4" id = "text-2-3-2" >
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< p >
Let’ s apply integral force feedback by integration \(F_m\):
\[ F = F^\prime - \frac{g}{s} F_m \]
< / p >
< p >
And we finally obtain:
\[ x = \frac{1}{ms^2 + mgs + k} F^\prime + \frac{1 + \frac{g}{s}}{ms^2 + mgs + k} F_d + \frac{k}{ms^2 + mgs + k} x_\mu \]
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > K_iff = 2< span class = "org-type" > *< / span > sqrt(k< span class = "org-type" > /< / span > m)< span class = "org-type" > /< / span > s;
K_iff.InputName = {< span class = "org-string" > 'Fm'< / span > };
K_iff.OutputName = {< span class = "org-string" > 'F'< / span > };
Gpz_iff = feedback(Gpz, K_iff, < span class = "org-string" > 'name'< / span > );
< / pre >
< / div >
< p >
Now let’ s consider that \(x_\mu = d_\mu + T_\mu r\)
< / p >
< p >
\[ x = \frac{1}{ms^2 + mgs + k} F^\prime + \frac{1 + \frac{g}{s}}{ms^2 + mgs + k} F_d + \frac{k}{ms^2 + mgs + k} d_\mu + \frac{T_\mu k}{ms^2 + mgs + k} r \]
< / p >
< p >
And \(\epsilon = r - x\):
\[ \epsilon = \frac{1}{ms^2 + mgs + k} F^\prime + \frac{1 + \frac{g}{s}}{ms^2 + mgs + k} F_d + \frac{k}{ms^2 + mgs + k} d_\mu + \frac{ms^2 + mgs + k - T_\mu k}{ms^2 + mgs + k} r \]
< / p >
< div class = "important" >
< p >
\[ \epsilon = \frac{1}{ms^2 + mgs + k} F^\prime + \frac{1 + \frac{g}{s}}{ms^2 + mgs + k} F_d + \frac{k}{ms^2 + mgs + k} d_\mu + \frac{ms^2 + mgs + S_\mu k}{ms^2 + mgs + k} r \]
< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-org5011ab0" class = "outline-4" >
< h4 id = "org5011ab0" > < span class = "section-number-4" > 2.3.3< / span > Low pass filter< / h4 >
< div class = "outline-text-4" id = "text-2-3-3" >
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< p >
Instead of a pure integrator, let’ s use a low pass filter with a cut-off frequency above the bandwidth of the micro-station \(\omega_mu\)
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-comment" > % K_iff = (2*sqrt(k/m)/(2*wmu))*(1/(1 + s/(2*wmu)));< / span >
< span class = "org-comment" > % K_iff.InputName = {'Fm'};< / span >
< span class = "org-comment" > % K_iff.OutputName = {'F'};< / span >
< span class = "org-comment" > % Gpz_iff = feedback(Gpz, K_iff, 'name');< / span >
< / pre >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-org4fce174" class = "outline-3" >
< h3 id = "org4fce174" > < span class = "section-number-3" > 2.4< / span > Comparison< / h3 >
< div class = "outline-text-3" id = "text-2-4" >
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< div id = "orgc10daac" class = "figure" >
< p > < img src = "figs/comp_iff_dvf_simplified.png" alt = "comp_iff_dvf_simplified.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 6: < / span > Obtained transfer functions for DVF and IFF (< a href = "./figs/comp_iff_dvf_simplified.png" > png< / a > , < a href = "./figs/comp_iff_dvf_simplified.pdf" > pdf< / a > )< / p >
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< / div >
< table border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
< colgroup >
< col class = "org-left" / >
< col class = "org-left" / >
< col class = "org-left" / >
< col class = "org-left" / >
< / colgroup >
< thead >
< tr >
< th scope = "col" class = "org-left" >   < / th >
< th scope = "col" class = "org-left" > \(d_\mu\)< / th >
< th scope = "col" class = "org-left" > \(F_d\)< / th >
< th scope = "col" class = "org-left" > \(w\)< / th >
< / tr >
< / thead >
< tbody >
< tr >
< td class = "org-left" > IFF< / td >
< td class = "org-left" > Better filtering of the vibrations< / td >
< td class = "org-left" > More sensitive to External forces< / td >
< td class = "org-left" >   < / td >
< / tr >
< tr >
< td class = "org-left" > DVF< / td >
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< td class = "org-left" > Opposite< / td >
< td class = "org-left" > Opposite< / td >
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< td class = "org-left" > Little bit better at low frequencies< / td >
< / tr >
< / tbody >
< / table >
< / div >
< / div >
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< div id = "outline-container-org5e0585d" class = "outline-3" >
< h3 id = "org5e0585d" > < span class = "section-number-3" > 2.5< / span > Control using \(x\)< / h3 >
< div class = "outline-text-3" id = "text-2-5" >
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< / div >
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< div id = "outline-container-orgfab8395" class = "outline-4" >
< h4 id = "orgfab8395" > < span class = "section-number-4" > 2.5.1< / span > Analytical analysis< / h4 >
< div class = "outline-text-4" id = "text-2-5-1" >
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< p >
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Let’ s first consider that only the output \(x\) is used for feedback (Figure < a href = "#orgd366408" > 7< / a > )
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< / p >
< div id = "orgd366408" class = "figure" >
< p > < img src = "figs/nano_station_control_x.png" alt = "nano_station_control_x.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 7: < / span > Feedback diagram using \(x\)< / p >
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< / div >
< p >
We then have:
\[ \epsilon & = r - G_{\frac{x}{F}} K \epsilon - G_{\frac{x}{F_d}} F_d - G_{\frac{x}{x_\mu}} d_\mu - G_{\frac{x}{x_\mu}} T_\mu r \]
< / p >
< p >
And then:
< / p >
< div class = "important" >
< p >
\[ \epsilon = \frac{-G_{\frac{x}{F_d}}}{1 + G_{\frac{x}{F}}K} F_d + \frac{-G_{\frac{x}{x_\mu}}}{1 + G_{\frac{x}{F}}K} d_\mu + \frac{1 - G_{\frac{x}{x_\mu}} T_\mu}{1 + G_{\frac{x}{F}}K} r \]
< / p >
< / div >
< p >
With \(S = \frac{1}{1 + G_{\frac{x}{F}} K}\), we have:
\[ \epsilon = - S G_{\frac{x}{F_d}} F_d - S G_{\frac{x}{x_\mu}} d_\mu + S (1 - G_{\frac{x}{x_\mu}} T_\mu) r \]
< / p >
< p >
We have 3 terms that we would like to have small by design:
< / p >
< ul class = "org-ul" >
< li > \(G_{\frac{x}{F_d}} = \frac{1}{ms^2 + k}\): thus \(k\) and \(m\) should be high to lower the effect of direct forces \(F_d\)< / li >
< li > \(G_{\frac{x}{x_\mu}} = \frac{k}{ms^2 + k} = \frac{1}{1 + \frac{s^2}{\omega_\nu^2}}\): \(\omega_\nu\) should be small enough such that it filters out the vibrations of the micro-station< / li >
< li > \(1 - G_{\frac{x}{x_\mu}} T_\mu\)< / li >
< / ul >
< p >
\[ 1 - G_{\frac{x}{x_\mu}} T_\mu = 1 - \frac{1}{1 + \frac{s^2}{\omega_\nu^2}} T_\mu \]
< / p >
< p >
We can approximate \(T_\mu \approx \frac{1}{1 + \frac{s}{\omega_\mu}}\) to have:
< / p >
\begin{align*}
1 - G_{\frac{x}{x_\mu}} T_\mu & = 1 - \frac{1}{1 + \frac{s^2}{\omega_\nu^2}} \frac{1}{1 + \frac{s}{\omega_\mu}} \\
& \approx \frac{\frac{s}{\omega_\mu}}{1 + \frac{s}{\omega_\mu}} = S_\mu \text{ if } \omega_\nu > \omega_\mu \\
& \approx \frac{\frac{s^2}{\omega_\nu^2}}{1 + \frac{s^2}{\omega_\nu^2}} = \text{ if } \omega_\nu < \omega_\mu
\end{align*}
< p >
In our case, we have \(\omega_\nu > \omega_\mu\) and thus we cannot lower this term.
< / p >
< p >
Some implications on the design are summarized on table < a href = "#orga5207fc" > 2< / a > .
< / p >
< table id = "orga5207fc" border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
< caption class = "t-above" > < span class = "table-number" > Table 2:< / span > Design recommendation< / caption >
< colgroup >
< col class = "org-left" / >
< col class = "org-left" / >
< / colgroup >
< thead >
< tr >
< th scope = "col" class = "org-left" > Exogenous Outputs< / th >
< th scope = "col" class = "org-left" > Design recommendation< / th >
< / tr >
< / thead >
< tbody >
< tr >
< td class = "org-left" > \(F_d\)< / td >
< td class = "org-left" > high \(k\), high \(m\)< / td >
< / tr >
< tr >
< td class = "org-left" > \(d_\mu\)< / td >
< td class = "org-left" > low \(k\), high \(m\)< / td >
< / tr >
< tr >
< td class = "org-left" > \(r\)< / td >
< td class = "org-left" > no influence if \(\omega_\nu > \omega_\mu\)< / td >
< / tr >
< / tbody >
< / table >
< / div >
< / div >
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< div id = "outline-container-org625e3c2" class = "outline-4" >
< h4 id = "org625e3c2" > < span class = "section-number-4" > 2.5.2< / span > Control implementation< / h4 >
< div class = "outline-text-4" id = "text-2-5-2" >
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< p >
Controller for the damped plant using DVF.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > wb = 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > 50; < span class = "org-comment" > % control bandwidth [rad/s]< / span >
< span class = "org-comment" > % Lead< / span >
h = 2.0;
wz = wb< span class = "org-type" > /< / span > h; < span class = "org-comment" > % [rad/s]< / span >
wp = wb< span class = "org-type" > *< / span > h; < span class = "org-comment" > % [rad/s]< / span >
H = 1< span class = "org-type" > /< / span > h< span class = "org-type" > *< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > wz)< span class = "org-type" > /< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > wp);
< span class = "org-comment" > % Integrator until 10Hz< / span >
Hi = (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > 2< span class = "org-type" > /< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > /< / span > 10)< span class = "org-type" > /< / span > (s< span class = "org-type" > /< / span > 2< span class = "org-type" > /< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > /< / span > 10);
K = Hi< span class = "org-type" > *< / span > H< span class = "org-type" > *< / span > (1< span class = "org-type" > /< / span > s);
Kpz_dvf = K< span class = "org-type" > /< / span > abs(freqresp(K< span class = "org-type" > *< / span > Gpz_dvf(< span class = "org-string" > 'y'< / span > , < span class = "org-string" > 'F'< / span > ), wb));
Kpz_dvf.InputName = {< span class = "org-string" > 'e'< / span > };
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Kpz_dvf.OutputName = {< span class = "org-string" > 'Fi'< / span > };
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< / pre >
< / div >
< p >
Controller for the damped plant using IFF.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > wb = 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > 50; < span class = "org-comment" > % control bandwidth [rad/s]< / span >
< span class = "org-comment" > % Lead< / span >
h = 2.0;
wz = wb< span class = "org-type" > /< / span > h; < span class = "org-comment" > % [rad/s]< / span >
wp = wb< span class = "org-type" > *< / span > h; < span class = "org-comment" > % [rad/s]< / span >
H = 1< span class = "org-type" > /< / span > h< span class = "org-type" > *< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > wz)< span class = "org-type" > /< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > wp);
< span class = "org-comment" > % Integrator until 10Hz< / span >
Hi = (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > 2< span class = "org-type" > /< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > /< / span > 10)< span class = "org-type" > /< / span > (s< span class = "org-type" > /< / span > 2< span class = "org-type" > /< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > /< / span > 10);
K = Hi< span class = "org-type" > *< / span > H< span class = "org-type" > *< / span > (1< span class = "org-type" > /< / span > s);
Kpz_iff = K< span class = "org-type" > /< / span > abs(freqresp(K< span class = "org-type" > *< / span > Gpz_iff(< span class = "org-string" > 'y'< / span > , < span class = "org-string" > 'F'< / span > ), wb));
Kpz_iff.InputName = {< span class = "org-string" > 'e'< / span > };
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Kpz_iff.OutputName = {< span class = "org-string" > 'Fi'< / span > };
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< / pre >
< / div >
< p >
Loop gain
< / p >
< div id = "org0d0fb80" class = "figure" >
< p > < img src = "figs/simple_loop_gain_pz.png" alt = "simple_loop_gain_pz.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 8: < / span > Loop Gain (< a href = "./figs/simple_loop_gain_pz.png" > png< / a > , < a href = "./figs/simple_loop_gain_pz.pdf" > pdf< / a > )< / p >
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< / div >
< p >
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Let’ s connect all the systems as shown in Figure < a href = "#orgd366408" > 7< / a > .
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< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Sfb = sumblk(< span class = "org-string" > 'e = r2 - y'< / span > );
R = [tf(1); tf(1)];
R.InputName = {< span class = "org-string" > 'r'< / span > };
R.OutputName = {< span class = "org-string" > 'r1'< / span > , < span class = "org-string" > 'r2'< / span > };
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F = [tf(1); tf(1)];
F.InputName = {< span class = "org-string" > 'Fi'< / span > };
F.OutputName = {< span class = "org-string" > 'F'< / span > , < span class = "org-string" > 'Fu'< / span > };
Gpz_fb_dvf = connect(Gpz_dvf, Kpz_dvf, R, Sfb, F, {< span class = "org-string" > 'r'< / span > , < span class = "org-string" > 'dmu'< / span > , < span class = "org-string" > 'Fd'< / span > , < span class = "org-string" > 'w'< / span > }, {< span class = "org-string" > 'y'< / span > , < span class = "org-string" > 'e'< / span > , < span class = "org-string" > 'Fm'< / span > , < span class = "org-string" > 'd'< / span > , < span class = "org-string" > 'Fu'< / span > });
Gpz_fb_iff = connect(Gpz_iff, Kpz_iff, R, Sfb, F, {< span class = "org-string" > 'r'< / span > , < span class = "org-string" > 'dmu'< / span > , < span class = "org-string" > 'Fd'< / span > , < span class = "org-string" > 'w'< / span > }, {< span class = "org-string" > 'y'< / span > , < span class = "org-string" > 'e'< / span > , < span class = "org-string" > 'Fm'< / span > , < span class = "org-string" > 'd'< / span > , < span class = "org-string" > 'Fu'< / span > });
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org3db77f5" class = "outline-4" >
< h4 id = "org3db77f5" > < span class = "section-number-4" > 2.5.3< / span > Results< / h4 >
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< div class = "outline-text-4" id = "text-2-5-3" >
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< div id = "org2b4e783" class = "figure" >
< p > < img src = "figs/simple_hac_lac_results.png" alt = "simple_hac_lac_results.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 9: < / span > Obtained closed-loop transfer functions (< a href = "./figs/simple_hac_lac_results.png" > png< / a > , < a href = "./figs/simple_hac_lac_results.pdf" > pdf< / a > )< / p >
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< / div >
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< table border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
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< colgroup >
< col class = "org-left" / >
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< col class = "org-left" / >
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< col class = "org-left" / >
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< col class = "org-left" / >
< / colgroup >
< thead >
< tr >
< th scope = "col" class = "org-left" >   < / th >
< th scope = "col" class = "org-left" > Reference Tracking< / th >
< th scope = "col" class = "org-left" > Vibration Filtering< / th >
< th scope = "col" class = "org-left" > Compliance< / th >
< / tr >
< / thead >
< tbody >
< tr >
< td class = "org-left" > DVF< / td >
< td class = "org-left" > Similar behavior< / td >
< td class = "org-left" >   < / td >
< td class = "org-left" > Better for \(\omega < \omega_\nu\)< / td >
< / tr >
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< tr >
< td class = "org-left" > IFF< / td >
< td class = "org-left" > Similar behavior< / td >
< td class = "org-left" > Better for \(\omega > \omega_\nu\)< / td >
< td class = "org-left" >   < / td >
< / tr >
< / tbody >
< / table >
< / div >
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< / div >
< / div >
< / div >
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< div id = "outline-container-org7c4b4fc" class = "outline-2" >
< h2 id = "org7c4b4fc" > < span class = "section-number-2" > 3< / span > Comparison with the use of a Soft nano-hexapod< / h2 >
< div class = "outline-text-2" id = "text-3" >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > m = 50; < span class = "org-comment" > % [kg]< / span >
k = 1e3; < span class = "org-comment" > % [N/m]< / span >
Gn = 1< span class = "org-type" > /< / span > (m< span class = "org-type" > *< / span > s< span class = "org-type" > ^< / span > 2 < span class = "org-type" > +< / span > k)< span class = "org-type" > *< / span > [< span class = "org-type" > -< / span > k, k< span class = "org-type" > *< / span > m< span class = "org-type" > *< / span > s< span class = "org-type" > ^< / span > 2, m< span class = "org-type" > *< / span > s< span class = "org-type" > ^< / span > 2; 1, < span class = "org-type" > -< / span > m< span class = "org-type" > *< / span > s< span class = "org-type" > ^< / span > 2, 1; 1, k, 1];
Gn.InputName = {< span class = "org-string" > 'Fd'< / span > , < span class = "org-string" > 'xmu'< / span > , < span class = "org-string" > 'F'< / span > };
Gn.OutputName = {< span class = "org-string" > 'Fm'< / span > , < span class = "org-string" > 'd'< / span > , < span class = "org-string" > 'x'< / span > };
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > wmu = 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > 50; < span class = "org-comment" > % [rad/s]< / span >
Tmu = 1< span class = "org-type" > /< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > wmu);
Tmu.InputName = {< span class = "org-string" > 'r1'< / span > };
Tmu.OutputName = {< span class = "org-string" > 'ymu'< / span > };
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > w0 = 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > 40;
xi = 0.5;
Tw = 1< span class = "org-type" > /< / span > (1 < span class = "org-type" > +< / span > 2< span class = "org-type" > *< / span > xi< span class = "org-type" > *< / span > s< span class = "org-type" > /< / span > w0 < span class = "org-type" > +< / span > s< span class = "org-type" > ^< / span > 2< span class = "org-type" > /< / span > w0< span class = "org-type" > ^< / span > 2);
Tw.InputName = {< span class = "org-string" > 'w1'< / span > };
Tw.OutputName = {< span class = "org-string" > 'dw'< / span > };
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Wsplit = [tf(1); tf(1)];
Wsplit.InputName = {< span class = "org-string" > 'w'< / span > };
Wsplit.OutputName = {< span class = "org-string" > 'w1'< / span > , < span class = "org-string" > 'w2'< / span > };
S = sumblk(< span class = "org-string" > 'xmu = ymu + dmu + dw'< / span > );
Sw = sumblk(< span class = "org-string" > 'y = x - w2'< / span > );
Gvc = connect(Gn, S, Wsplit, Tw, Tmu, Sw, {< span class = "org-string" > 'Fd'< / span > , < span class = "org-string" > 'dmu'< / span > , < span class = "org-string" > 'r1'< / span > , < span class = "org-string" > 'F'< / span > , < span class = "org-string" > 'w'< / span > }, {< span class = "org-string" > 'Fm'< / span > , < span class = "org-string" > 'd'< / span > , < span class = "org-string" > 'y'< / span > });
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Kvc_dvf = 2< span class = "org-type" > *< / span > sqrt(k< span class = "org-type" > *< / span > m)< span class = "org-type" > *< / span > s;
Kvc_dvf.InputName = {< span class = "org-string" > 'd'< / span > };
Kvc_dvf.OutputName = {< span class = "org-string" > 'F'< / span > };
Gvc_dvf = feedback(Gvc, Kvc_dvf, < span class = "org-string" > 'name'< / span > );
Kvc_iff = 2< span class = "org-type" > *< / span > sqrt(k< span class = "org-type" > /< / span > m)< span class = "org-type" > /< / span > s;
Kvc_iff.InputName = {< span class = "org-string" > 'Fm'< / span > };
Kvc_iff.OutputName = {< span class = "org-string" > 'F'< / span > };
Gvc_iff = feedback(Gvc, Kvc_iff, < span class = "org-string" > 'name'< / span > );
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > wb = 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > 100; < span class = "org-comment" > % control bandwidth [rad/s]< / span >
< span class = "org-comment" > % Lead< / span >
h = 2.0;
wz = wb< span class = "org-type" > /< / span > h; < span class = "org-comment" > % [rad/s]< / span >
wp = wb< span class = "org-type" > *< / span > h; < span class = "org-comment" > % [rad/s]< / span >
H = 1< span class = "org-type" > /< / span > h< span class = "org-type" > *< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > wz)< span class = "org-type" > /< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > wp);
Kvc_dvf = H< span class = "org-type" > /< / span > abs(freqresp(H< span class = "org-type" > *< / span > Gvc_dvf(< span class = "org-string" > 'y'< / span > , < span class = "org-string" > 'F'< / span > ), wb));
Kvc_dvf.InputName = {< span class = "org-string" > 'e'< / span > };
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Kvc_dvf.OutputName = {< span class = "org-string" > 'Fi'< / span > };
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Kvc_iff = H< span class = "org-type" > /< / span > abs(freqresp(H< span class = "org-type" > *< / span > Gvc_iff(< span class = "org-string" > 'y'< / span > , < span class = "org-string" > 'F'< / span > ), wb));
Kvc_iff.InputName = {< span class = "org-string" > 'e'< / span > };
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Kvc_iff.OutputName = {< span class = "org-string" > 'Fi'< / span > };
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< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Sfb = sumblk(< span class = "org-string" > 'e = r2 - y'< / span > );
R = [tf(1); tf(1)];
R.InputName = {< span class = "org-string" > 'r'< / span > };
R.OutputName = {< span class = "org-string" > 'r1'< / span > , < span class = "org-string" > 'r2'< / span > };
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F = [tf(1); tf(1)];
F.InputName = {< span class = "org-string" > 'Fi'< / span > };
F.OutputName = {< span class = "org-string" > 'F'< / span > , < span class = "org-string" > 'Fu'< / span > };
Gvc_fb_dvf = connect(Gvc_dvf, Kvc_dvf, R, Sfb, F, {< span class = "org-string" > 'r'< / span > , < span class = "org-string" > 'dmu'< / span > , < span class = "org-string" > 'Fd'< / span > , < span class = "org-string" > 'w'< / span > }, {< span class = "org-string" > 'y'< / span > , < span class = "org-string" > 'e'< / span > , < span class = "org-string" > 'Fm'< / span > , < span class = "org-string" > 'd'< / span > , < span class = "org-string" > 'Fu'< / span > });
Gvc_fb_iff = connect(Gvc_iff, Kvc_iff, R, Sfb, F, {< span class = "org-string" > 'r'< / span > , < span class = "org-string" > 'dmu'< / span > , < span class = "org-string" > 'Fd'< / span > , < span class = "org-string" > 'w'< / span > }, {< span class = "org-string" > 'y'< / span > , < span class = "org-string" > 'e'< / span > , < span class = "org-string" > 'Fm'< / span > , < span class = "org-string" > 'd'< / span > , < span class = "org-string" > 'Fu'< / span > });
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< / pre >
< / div >
< div id = "org3817d8a" class = "figure" >
< p > < img src = "figs/simple_hac_lac_results_soft.png" alt = "simple_hac_lac_results_soft.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 10: < / span > Obtained closed-loop transfer functions (< a href = "./figs/simple_hac_lac_results_soft.png" > png< / a > , < a href = "./figs/simple_hac_lac_results_soft.pdf" > pdf< / a > )< / p >
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< / div >
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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" / >
< col class = "org-left" / >
< / colgroup >
< thead >
< tr >
< th scope = "col" class = "org-left" >   < / th >
< th scope = "col" class = "org-left" > Reference Tracking< / th >
< th scope = "col" class = "org-left" > Vibration Filtering< / th >
< th scope = "col" class = "org-left" > Compliance< / th >
< / tr >
< / thead >
< tbody >
< tr >
< td class = "org-left" > DVF< / td >
< td class = "org-left" > Similar behavior< / td >
< td class = "org-left" >   < / td >
< td class = "org-left" > Better for \(\omega < \omega_\nu\)< / td >
< / tr >
< tr >
< td class = "org-left" > IFF< / td >
< td class = "org-left" > Similar behavior< / td >
< td class = "org-left" > Better for \(\omega > \omega_\nu\)< / td >
< td class = "org-left" >   < / td >
< / tr >
< / tbody >
< / table >
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< div id = "org55e0fe2" class = "figure" >
< p > < img src = "figs/simple_comp_vc_pz.png" alt = "simple_comp_vc_pz.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 11: < / span > Comparison of the closed-loop transfer functions for Soft and Stiff nano-hexapod (< a href = "./figs/simple_comp_vc_pz.png" > png< / a > , < a href = "./figs/simple_comp_vc_pz.pdf" > pdf< / a > )< / p >
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< / div >
< table border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
< colgroup >
< col class = "org-left" / >
< col class = "org-center" / >
< col class = "org-center" / >
< / colgroup >
< thead >
< tr >
< th scope = "col" class = "org-left" >   < / th >
< th scope = "col" class = "org-center" > < b > Soft< / b > < / th >
< th scope = "col" class = "org-center" > < b > Stiff< / b > < / th >
< / tr >
< / thead >
< tbody >
< tr >
< td class = "org-left" > < b > Reference Tracking< / b > < / td >
< td class = "org-center" > =< / td >
< td class = "org-center" > =< / td >
< / tr >
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< tr >
< td class = "org-left" > < b > Ground Motion< / b > < / td >
< td class = "org-center" > =< / td >
< td class = "org-center" > =< / td >
< / tr >
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< tr >
< td class = "org-left" > < b > Vibration Isolation< / b > < / td >
< td class = "org-center" > +< / td >
< td class = "org-center" > -< / td >
< / tr >
< tr >
< td class = "org-left" > < b > Compliance< / b > < / td >
< td class = "org-center" > -< / td >
< td class = "org-center" > +< / td >
< / tr >
< / tbody >
< / table >
< / div >
< / div >
< div id = "outline-container-orgc0253c3" class = "outline-2" >
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< h2 id = "orgc0253c3" > < span class = "section-number-2" > 4< / span > Estimate the level of vibration< / h2 >
< div class = "outline-text-2" id = "text-4" >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > gm = load(< span class = "org-string" > './mat/psd_gm.mat'< / span > , < span class = "org-string" > 'f'< / span > , < span class = "org-string" > 'psd_gm'< / span > );
rz = load(< span class = "org-string" > './mat/pxsp_r.mat'< / span > , < span class = "org-string" > 'f'< / span > , < span class = "org-string" > 'pxsp_r'< / span > );
tyz = load(< span class = "org-string" > './mat/pxz_ty_r.mat'< / span > , < span class = "org-string" > 'f'< / span > , < span class = "org-string" > 'pxz_ty_r'< / span > );
< / pre >
< / div >
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< p >
If we note the PSD \(\Gamma\):
\[ \Gamma_y = |G_{\frac{y}{w}}|^2 \Gamma_w + |G_{\frac{y}{x_\mu}}|^2 \Gamma_{x_\mu} \]
< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > x_pz = abs(squeeze(freqresp(Gpz_fb_iff(< span class = "org-string" > 'y'< / span > , < span class = "org-string" > 'dmu'< / span > ), f, < span class = "org-string" > 'Hz'< / span > )))< span class = "org-type" > .^< / span > 2< span class = "org-type" > .*< / span > (psd_rz < span class = "org-type" > +< / span > psd_ty) < span class = "org-type" > +< / span > abs(squeeze(freqresp(Gpz_fb_iff(< span class = "org-string" > 'y'< / span > , < span class = "org-string" > 'w'< / span > ), f, < span class = "org-string" > 'Hz'< / span > )))< span class = "org-type" > .^< / span > 2< span class = "org-type" > .*< / span > (psd_gm);
x_vc = abs(squeeze(freqresp(Gvc_fb_iff(< span class = "org-string" > 'y'< / span > , < span class = "org-string" > 'dmu'< / span > ), f, < span class = "org-string" > 'Hz'< / span > )))< span class = "org-type" > .^< / span > 2< span class = "org-type" > .*< / span > (psd_rz < span class = "org-type" > +< / span > psd_ty) < span class = "org-type" > +< / span > abs(squeeze(freqresp(Gvc_fb_iff(< span class = "org-string" > 'y'< / span > , < span class = "org-string" > 'w'< / span > ), f, < span class = "org-string" > 'Hz'< / span > )))< span class = "org-type" > .^< / span > 2< span class = "org-type" > .*< / span > (psd_gm);
2020-03-13 17:40:22 +01:00
< / pre >
< / div >
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< div id = "org858053c" class = "figure" >
< p > < img src = "figs/simple_asd_motion_error.png" alt = "simple_asd_motion_error.png" / >
< / p >
< p > < span class = "figure-number" > Figure 12: < / span > ASD of the position error due to Ground Motion and Vibration (< a href = "./figs/simple_asd_motion_error.png" > png< / a > , < a href = "./figs/simple_asd_motion_error.pdf" > pdf< / a > )< / p >
< / div >
< p >
Actuator usage
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > F_pz = abs(squeeze(freqresp(Gpz_fb_iff(< span class = "org-string" > 'Fu'< / span > , < span class = "org-string" > 'dmu'< / span > ), f, < span class = "org-string" > 'Hz'< / span > )))< span class = "org-type" > .^< / span > 2< span class = "org-type" > .*< / span > (psd_rz < span class = "org-type" > +< / span > psd_ty) < span class = "org-type" > +< / span > abs(squeeze(freqresp(Gpz_fb_iff(< span class = "org-string" > 'Fu'< / span > , < span class = "org-string" > 'w'< / span > ), f, < span class = "org-string" > 'Hz'< / span > )))< span class = "org-type" > .^< / span > 2< span class = "org-type" > .*< / span > (psd_gm);
F_vc = abs(squeeze(freqresp(Gvc_fb_iff(< span class = "org-string" > 'Fu'< / span > , < span class = "org-string" > 'dmu'< / span > ), f, < span class = "org-string" > 'Hz'< / span > )))< span class = "org-type" > .^< / span > 2< span class = "org-type" > .*< / span > (psd_rz < span class = "org-type" > +< / span > psd_ty) < span class = "org-type" > +< / span > abs(squeeze(freqresp(Gvc_fb_iff(< span class = "org-string" > 'Fu'< / span > , < span class = "org-string" > 'w'< / span > ), f, < span class = "org-string" > 'Hz'< / span > )))< span class = "org-type" > .^< / span > 2< span class = "org-type" > .*< / span > (psd_gm);
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > sqrt(trapz(f, F_pz))
sqrt(trapz(f, F_vc))
< / pre >
< / div >
< pre class = "example" >
sqrt(trapz(f, F_pz))
ans =
84.8961762069446
sqrt(trapz(f, F_vc))
ans =
0.0387785981815527
< / pre >
2020-03-13 17:40:22 +01:00
< / div >
< / div >
< div id = "outline-container-org764c4a9" class = "outline-2" >
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< h2 id = "org764c4a9" > < span class = "section-number-2" > 5< / span > Requirements on the norm of closed-loop transfer functions< / h2 >
< div class = "outline-text-2" id = "text-5" >
< / div >
< div id = "outline-container-org27379f3" class = "outline-3" >
< h3 id = "org27379f3" > < span class = "section-number-3" > 5.1< / span > Approximation of the ASD of perturbations< / h3 >
< div class = "outline-text-3" id = "text-5-1" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > G_rz = 1e< span class = "org-type" > -< / span > 9< span class = "org-type" > *< / span > 1< span class = "org-type" > /< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > 2< span class = "org-type" > /< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > /< / span > 0.5)< span class = "org-type" > ^< / span > 2< span class = "org-type" > *< / span > (s < span class = "org-type" > +< / span > 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > 1)< span class = "org-type" > *< / span > (s < span class = "org-type" > +< / span > 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > 10)< span class = "org-type" > *< / span > (1< span class = "org-type" > /< / span > ((1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > 2< span class = "org-type" > /< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > /< / span > 100)< span class = "org-type" > ^< / span > 2));
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > G_gm = 1e< span class = "org-type" > -< / span > 8< span class = "org-type" > *< / span > 1< span class = "org-type" > /< / span > s< span class = "org-type" > ^< / span > 2< span class = "org-type" > *< / span > (s < span class = "org-type" > +< / span > 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > 1)< span class = "org-type" > ^< / span > 2< span class = "org-type" > *< / span > (1< span class = "org-type" > /< / span > ((1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > 2< span class = "org-type" > /< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > /< / span > 10)< span class = "org-type" > ^< / span > 3));
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-orgff3d823" class = "outline-3" >
< h3 id = "orgff3d823" > < span class = "section-number-3" > 5.2< / span > Wanted ASD of outputs< / h3 >
< div class = "outline-text-3" id = "text-5-2" >
< p >
Wanted ASD of motion error
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > y_wanted = 100e< span class = "org-type" > -< / span > 9; < span class = "org-comment" > % 10nm rms wanted< / span >
y_bw = 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > 100; < span class = "org-comment" > % bandwidth [rad/s]< / span >
G_y = 2< span class = "org-type" > *< / span > y_wanted< span class = "org-type" > /< / span > sqrt(y_bw) < span class = "org-type" > *< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > y_bw< span class = "org-type" > /< / span > 10) < span class = "org-type" > /< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > y_bw);
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > sqrt(trapz(f, abs(squeeze(freqresp(G_y, f, < span class = "org-string" > 'Hz'< / span > )))< span class = "org-type" > .^< / span > 2))
< / pre >
< / div >
< pre class = "example" >
sqrt(trapz(f, abs(squeeze(freqresp(G_y, f, 'Hz'))).^2))
ans =
9.47118350214793e-08
< / pre >
< / div >
< / div >
< div id = "outline-container-org8c6b37c" class = "outline-3" >
< h3 id = "org8c6b37c" > < span class = "section-number-3" > 5.3< / span > Limiting the bandwidth< / h3 >
< div class = "outline-text-3" id = "text-5-3" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > wF = 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > 10;
G_F = 100000< span class = "org-type" > *< / span > (wF < span class = "org-type" > +< / span > s)< span class = "org-type" > ^< / span > 2;
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-org50054f2" class = "outline-3" >
< h3 id = "org50054f2" > < span class = "section-number-3" > 5.4< / span > Generalized Weighted plant< / h3 >
< div class = "outline-text-3" id = "text-5-4" >
< p >
Let’ s create a generalized weighted plant for controller synthesis.
< / p >
< p >
Let’ s start simple:
< / p >
< 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" >   < / th >
< th scope = "col" class = "org-left" > < b > Symbol< / b > < / th >
< th scope = "col" class = "org-left" > < b > Meaning< / b > < / th >
< / tr >
< / thead >
< tbody >
< tr >
< td class = "org-left" > < b > Exogenous Inputs< / b > < / td >
< td class = "org-left" > \(x_\mu\)< / td >
< td class = "org-left" > Motion of the $ν $-hexapod’ s base< / td >
< / tr >
< / tbody >
< tbody >
< tr >
< td class = "org-left" > < b > Exogenous Outputs< / b > < / td >
< td class = "org-left" > \(y\)< / td >
< td class = "org-left" > Motion error of the Payload< / td >
< / tr >
< / tbody >
< tbody >
< tr >
< td class = "org-left" > < b > Sensed Outputs< / b > < / td >
< td class = "org-left" > \(y\)< / td >
< td class = "org-left" > Motion error of the Payload< / td >
< / tr >
< / tbody >
< tbody >
< tr >
< td class = "org-left" > < b > Control Signals< / b > < / td >
< td class = "org-left" > \(F\)< / td >
< td class = "org-left" > Actuator Inputs< / td >
< / tr >
< / tbody >
< / table >
< p >
Add \(F\) as output.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > F = [tf(1); tf(1)];
F.InputName = {< span class = "org-string" > 'Fi'< / span > };
F.OutputName = {< span class = "org-string" > 'F'< / span > , < span class = "org-string" > 'Fu'< / span > };
P_pz = connect(F, Gpz_dvf, {< span class = "org-string" > 'dmu'< / span > , < span class = "org-string" > 'Fi'< / span > }, {< span class = "org-string" > 'y'< / span > , < span class = "org-string" > 'Fu'< / span > , < span class = "org-string" > 'y'< / span > })
P_vc = connect(F, Gvc_dvf, {< span class = "org-string" > 'dmu'< / span > , < span class = "org-string" > 'Fi'< / span > }, {< span class = "org-string" > 'y'< / span > , < span class = "org-string" > 'Fu'< / span > , < span class = "org-string" > 'y'< / span > })
< / pre >
< / div >
< p >
Normalization.
< / p >
< p >
We multiply the plant input by \(G_{rz}\) and the plant output by \(G_y^{-1}\):
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > P_pz_norm = blkdiag(inv(G_y), inv(G_F), 1)< span class = "org-type" > *< / span > P_pz< span class = "org-type" > *< / span > blkdiag(G_rz, 1);
P_pz_norm.OutputName = {< span class = "org-string" > 'z'< / span > , < span class = "org-string" > 'F'< / span > , < span class = "org-string" > 'y'< / span > };
P_pz_norm.InputName = {< span class = "org-string" > 'w'< / span > , < span class = "org-string" > 'u'< / span > };
P_vc_norm = blkdiag(inv(G_y), inv(G_F), 1)< span class = "org-type" > *< / span > P_vc< span class = "org-type" > *< / span > blkdiag(G_rz, 1);
P_vc_norm.OutputName = {< span class = "org-string" > 'z'< / span > , < span class = "org-string" > 'F'< / span > , < span class = "org-string" > 'y'< / span > };
P_vc_norm.InputName = {< span class = "org-string" > 'w'< / span > , < span class = "org-string" > 'u'< / span > };
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-org949ab66" class = "outline-3" >
< h3 id = "org949ab66" > < span class = "section-number-3" > 5.5< / span > Synthesis< / h3 >
< div class = "outline-text-3" id = "text-5-5" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > [Kpz_dvf,CL_vc,< span class = "org-type" > ~< / span > ] = hinfsyn(minreal(P_pz_norm), 1, 1, < span class = "org-string" > 'TOLGAM'< / span > , 0.001, < span class = "org-string" > 'METHOD'< / span > , < span class = "org-string" > 'LMI'< / span > , < span class = "org-string" > 'DISPLAY'< / span > , < span class = "org-string" > 'on'< / span > );
Kpz_dvf.InputName = {< span class = "org-string" > 'e'< / span > };
Kpz_dvf.OutputName = {< span class = "org-string" > 'Fi'< / span > };
[Kvc_dvf,CL_pz,< span class = "org-type" > ~< / span > ] = hinfsyn(minreal(P_vc_norm), 1, 1, < span class = "org-string" > 'TOLGAM'< / span > , 0.001, < span class = "org-string" > 'METHOD'< / span > , < span class = "org-string" > 'LMI'< / span > , < span class = "org-string" > 'DISPLAY'< / span > , < span class = "org-string" > 'on'< / span > );
Kvc_dvf.InputName = {< span class = "org-string" > 'e'< / span > };
Kvc_dvf.OutputName = {< span class = "org-string" > 'Fi'< / span > };
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-orgfe970e4" class = "outline-3" >
< h3 id = "orgfe970e4" > < span class = "section-number-3" > 5.6< / span > Loop Gain< / h3 >
< div class = "outline-text-3" id = "text-5-6" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Sfb = sumblk(< span class = "org-string" > 'e = r2 - y'< / span > );
R = [tf(1); tf(1)];
R.InputName = {< span class = "org-string" > 'r'< / span > };
R.OutputName = {< span class = "org-string" > 'r1'< / span > , < span class = "org-string" > 'r2'< / span > };
F = [tf(1); tf(1)];
F.InputName = {< span class = "org-string" > 'Fi'< / span > };
F.OutputName = {< span class = "org-string" > 'F'< / span > , < span class = "org-string" > 'Fu'< / span > };
Gpz_fb_dvf = connect(Gpz_dvf, < span class = "org-type" > -< / span > Kpz_dvf, R, Sfb, F, {< span class = "org-string" > 'r'< / span > , < span class = "org-string" > 'dmu'< / span > , < span class = "org-string" > 'Fd'< / span > , < span class = "org-string" > 'w'< / span > }, {< span class = "org-string" > 'y'< / span > , < span class = "org-string" > 'e'< / span > , < span class = "org-string" > 'Fm'< / span > , < span class = "org-string" > 'd'< / span > , < span class = "org-string" > 'Fu'< / span > });
Gvc_fb_dvf = connect(Gvc_dvf, < span class = "org-type" > -< / span > Kvc_dvf, R, Sfb, F, {< span class = "org-string" > 'r'< / span > , < span class = "org-string" > 'dmu'< / span > , < span class = "org-string" > 'Fd'< / span > , < span class = "org-string" > 'w'< / span > }, {< span class = "org-string" > 'y'< / span > , < span class = "org-string" > 'e'< / span > , < span class = "org-string" > 'Fm'< / span > , < span class = "org-string" > 'd'< / span > , < span class = "org-string" > 'Fu'< / span > });
< / pre >
< / div >
< / div >
< / div >
2020-03-26 17:25:43 +01:00
< div id = "outline-container-org5211e8c" class = "outline-3" >
< h3 id = "org5211e8c" > < span class = "section-number-3" > 5.7< / span > Results< / h3 >
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< / div >
< div id = "outline-container-orgb18d7df" class = "outline-3" >
< h3 id = "orgb18d7df" > < span class = "section-number-3" > 5.8< / span > Requirements< / h3 >
< div class = "outline-text-3" id = "text-5-8" >
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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 >
< tbody >
< tr >
< td class = "org-left" > reference tracking< / td >
< td class = "org-left" > \(\epsilon/r\)< / td >
< td class = "org-left" > -120dB at 1Hz< / td >
< / tr >
< tr >
< td class = "org-left" > vibration isolation< / td >
< td class = "org-left" > \(x/x_\mu\)< / td >
< td class = "org-left" > -60dB above 10Hz< / td >
< / tr >
< tr >
< td class = "org-left" > compliance< / td >
< td class = "org-left" > \(x/F_d\)< / td >
< td class = "org-left" >   < / td >
< / tr >
< / tbody >
< / table >
< / div >
< / div >
< / div >
2020-03-17 11:23:47 +01:00
< / div >
2020-03-13 17:40:22 +01:00
< div id = "postamble" class = "status" >
< p class = "author" > Author: Dehaeze Thomas< / p >
2020-03-26 17:25:43 +01:00
< p class = "date" > Created: 2020-03-26 jeu. 17:25< / p >
2020-03-13 17:40:22 +01:00
< / div >
< / body >
< / html >
