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Thomas Dehaeze
2020-03-26 17:25:43 +01:00
parent de8ba392ed
commit 8c6768f372
19 changed files with 1712 additions and 332 deletions
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4
org/control_active_damping.org
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@@ -87,7 +87,7 @@ After that, a tomography experiment is simulation without any active damping tec
<<matlab-init>>
#+end_src
#+begin_src matlab :tangle no
#+begin_src matlab
simulinkproject('../');
#+end_src
@@ -2985,7 +2985,7 @@ Inertial Control should not be used.
#+end_src
#+begin_src matlab
cd('../');
simulinkproject('../');
#+end_src
** Load the plants

4
org/control_decentralized.org
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@@ -99,10 +99,6 @@ The signals are:
\draw[->] (Ex.east) -- (J.west) node[above left]{$\bm{r}_{\mathcal{X}_n}$};
\draw[->] (J.east) -- (subr.west) node[above left]{$\bm{r}_{\mathcal{L}}$};
\draw[<-] (Ex.west)node[above left]{$\bm{r}_{\mathcal{X}}$} -- ++(-1, 0);
% \draw[->] (J.east) -- (subr.west) node[above left]{$\bm{r}_{\mathcal{L}}$};
% \draw[->] (subr.east) -- (K.west) node[above left]{$\bm{\epsilon}_\mathcal{L}$};
% \draw[->] (G.east) node[above right]{$\bm{\mathcal{L}}$} -| ($(G.east)+(1, -1)$) -| (subr.south);
\end{tikzpicture}
#+end_src

2
org/control_requirements.org
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@@ -554,7 +554,7 @@ Instead of a pure integrator, let's use a low pass filter with a cut-off frequen
| | $d_\mu$ | $F_d$ | $w$ |
|-----+------------------------------------+-----------------------------------+--------------------------------------|
| IFF | Better filtering of the vibrations | More sensitive to External forces | |
| DVF | inverse | inverse | Little bit better at low frequencies |
| DVF | Opposite | Opposite | Little bit better at low frequencies |
** Control using $x$
*** Analytical analysis

17
org/uncertainty_payload.org
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@@ -350,13 +350,13 @@ The obtained dynamics from $F$ to $x$ for the three isolation platform are shown
* Generalization to arbitrary dynamics
<<sec:arbitrary_dynamics>>
** Introduction
Let's now consider a general payload described by its *impedance* $G^\prime(s) = \frac{x}{F^\prime}$ as shown in Figure [[fig:general_payload_impdeance]].
Let's now consider a general payload described by its *impedance* $G^\prime(s) = \frac{x}{F^\prime}$ as shown in Figure [[fig:general_payload_impedance]].
#+begin_note
Note here that we use the term /impedance/, however, the mechanical impedance is usually defined as the ratio of the velocity over the force $\dot{x}/F^\prime$. We should refer to /resistance/ instead of /impedance/.
#+end_note
#+begin_src latex :file general_payload_impdeance.pdf
#+begin_src latex :file general_payload_impedance.pdf
\begin{tikzpicture}
\def\massw{2.2} % Width of the masses
\def\massh{0.8} % Height of the masses
@@ -379,10 +379,10 @@ Note here that we use the term /impedance/, however, the mechanical impedance is
\end{tikzpicture}
#+end_src
#+name: fig:general_support_compliance
#+name: fig:general_payload_impedance
#+caption: General support
#+RESULTS:
[[file:figs/general_payload_impdeance.png]]
[[file:figs/general_payload_impedance.png]]
Now let's consider the system consisting of a mass-spring-system (the isolation platform) supporting the general payload as shown in Figure [[fig:general_payload_with_isolator]].
#+begin_src latex :file general_payload_with_isolator.pdf
@@ -441,8 +441,9 @@ We have to following equations of motion:
And by eliminating $F^\prime$, we find the plant dynamics $G(s) = \frac{x}{F}$.
#+begin_important
#+name: eq:plant_dynamics_general_payload
\begin{equation}
\frac{x}{F} = \frac{1}{ms^2 + cs + k + G^\prime(s)} \label{eq:plant_dynamics_general_payload}
\frac{x}{F} = \frac{1}{ms^2 + cs + k + G^\prime(s)}
\end{equation}
#+end_important
@@ -467,8 +468,9 @@ In order to verify that the formula is correct, let's take the same mass-spring-
By eliminating $x^\prime$ of the equations, we obtain:
#+begin_important
#+name: eq:impedance_mass_spring_damper
\begin{equation}
G^\prime(s) = \frac{F^\prime}{x} = \frac{m^\prime s^2 (c^\prime s + k)}{m^\prime s^2 + c^\prime s + k^\prime} \label{eq:impedance_mass_spring_damper}
G^\prime(s) = \frac{F^\prime}{x} = \frac{m^\prime s^2 (c^\prime s + k)}{m^\prime s^2 + c^\prime s + k^\prime}
\end{equation}
The impedance of a 1dof mass-spring-damper system is described by Eq. [[eq:impedance_mass_spring_damper]].
@@ -667,7 +669,8 @@ Stiff Isolation Platform
G_stiff = 1/(m*s^2 + c_stiff*s + k_stiff + Gp);
#+end_src
The obtained transfer functions $x/F$ for each of the three platforms are shown in Figure [[fig:plant_uncertainty_stiffness_isolator]].
The obtained transfer functions $x/F$ for each of the three platforms are shown in Figure [[fig:plant_uncertainty_impedance_payload]].
#+begin_src matlab :exports none
freqs = logspace(0, 3, 1000);

45
org/uncertainty_support.org
View File

@@ -272,7 +272,7 @@ The obtained dynamics from $F$ to $x$ for the three isolation platform are shown
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]);
ylabel('Magnitude [m/N]');
title('Soft Platform');
title('$\omega_0 \ll \omega_0^\prime$');
hold off;
ax4 = subplot(2,3,4);
@@ -293,7 +293,7 @@ The obtained dynamics from $F$ to $x$ for the three isolation platform are shown
end
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]);
title('Medium Stiff Platform');
title('$\omega_0 \approx \omega_0^\prime$');
hold off;
ax5 = subplot(2,3,5);
@@ -314,7 +314,7 @@ The obtained dynamics from $F$ to $x$ for the three isolation platform are shown
end
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]);
title('Stiff Platform');
title('$\omega_0 \gg \omega_0^\prime$');
hold off;
ax6 = subplot(2,3,6);
@@ -344,7 +344,7 @@ The obtained dynamics from $F$ to $x$ for the three isolation platform are shown
** Conclusion
#+begin_important
The soft platform dynamics does not seems to depend on the dynamics of the support.
The soft platform dynamics does not seems to depend on the dynamics of the support nor to be affect by the dynamic uncertainty of the support.
#+end_important
* Generalization to arbitrary dynamics
@@ -615,6 +615,7 @@ Stiff Isolation Platform
#+end_src
The obtained transfer functions $x/F$ for each of the three platforms are shown in Figure [[fig:plant_uncertainty_stiffness_isolator]].
#+begin_src matlab :exports none
freqs = logspace(0, 3, 1000);
@@ -633,9 +634,9 @@ The obtained transfer functions $x/F$ for each of the three platforms are shown
set(gca, 'XTickLabel',[]);
ylabel('Magnitude [dB]');
hold off;
title('Soft Isolator');
title('$\omega_0 \ll \omega_0^\prime$');
ax2 = subplot(2,3,4);
ax4 = subplot(2,3,4);
hold on;
for i = 1:length(Gs_soft)
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs_soft(:, :, i), freqs, 'Hz'))), '-', 'color', [0, 0, 0, 0.2]);
@@ -646,10 +647,7 @@ The obtained transfer functions $x/F$ for each of the three platforms are shown
ylabel('Phase [deg]');
hold off;
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
ax1 = subplot(2,3,2);
ax2 = subplot(2,3,2);
hold on;
for i = 1:length(Gs_mid)
plot(freqs, abs(squeeze(freqresp(Gs_mid(:, :, i), freqs, 'Hz'))), '-', 'color', [0, 0, 0, 0.2]);
@@ -657,9 +655,9 @@ The obtained transfer functions $x/F$ for each of the three platforms are shown
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]);
hold off;
title('Medium Stiff Isolator');
title('$\omega_0 \approx \omega_0^\prime$');
ax2 = subplot(2,3,5);
ax5 = subplot(2,3,5);
hold on;
for i = 1:length(Gs_mid)
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs_mid(:, :, i), freqs, 'Hz'))), '-', 'color', [0, 0, 0, 0.2]);
@@ -670,10 +668,7 @@ The obtained transfer functions $x/F$ for each of the three platforms are shown
xlabel('Frequency [Hz]');
hold off;
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
ax1 = subplot(2,3,3);
ax3 = subplot(2,3,3);
hold on;
for i = 1:length(Gs_stiff)
plot(freqs, abs(squeeze(freqresp(Gs_stiff(:, :, i), freqs, 'Hz'))), '-', 'color', [0, 0, 0, 0.2]);
@@ -681,9 +676,9 @@ The obtained transfer functions $x/F$ for each of the three platforms are shown
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]);
hold off;
title('Stiff Isolator');
title('$\omega_0 \gg \omega_0^\prime$');
ax2 = subplot(2,3,6);
ax6 = subplot(2,3,6);
hold on;
for i = 1:length(Gs_stiff)
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs_stiff(:, :, i), freqs, 'Hz'))), '-', 'color', [0, 0, 0, 0.2]);
@@ -693,8 +688,10 @@ The obtained transfer functions $x/F$ for each of the three platforms are shown
ylim([-180 180]);
hold off;
linkaxes([ax1,ax2],'x');
linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'x');
xlim([freqs(1), freqs(end)]);
linkaxes([ax1,ax2,ax3],'y');
#+end_src
#+header: :tangle no :exports results :results none :noweb yes
@@ -717,8 +714,8 @@ Let's express the uncertainty of the plant $x/F$ as a function of the parameters
&= \frac{1}{ms^2 + cs + k + ms^2(cs + k)G_0^\prime(s)} \cdot \frac{1}{1 + \frac{ms^2(cs + k)G_0^\prime(s) w_I(s)}{ms^2 + cs + k + ms^2(cs + k)G_0^\prime(s)} \Delta(s)}\\
\end{align*}
We can rewrite that as an inverse multiplicative uncertainty (Figure [[fig:inverse_uncertainty_set]]):
#+begin_important
We can the plant dynamics that as an inverse multiplicative uncertainty (Figure [[fig:inverse_uncertainty_set]]):
\begin{equation}
\frac{x}{F} = G_0(s) (1 + w_{iI}(s) \Delta(s))^{-1}
\end{equation}
@@ -906,7 +903,7 @@ Let's fix $k = 10^7\ [N/m]$, $m = 100\ [kg]$ and see the evolution of $|w_{iI}(j
surf(freqs, xi, wiI_c_soft', 'FaceColor', 'interp', 'EdgeColor', 'none')
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Damping Ratio');
title('Soft Platform');
title('$\omega_0 \ll \omega_0^\prime$');
view([0 0 1]);
set(gca,'ColorScale','log')
colorbar('location', 'west');
@@ -916,7 +913,7 @@ Let's fix $k = 10^7\ [N/m]$, $m = 100\ [kg]$ and see the evolution of $|w_{iI}(j
surf(freqs, xi, wiI_c_mid', 'FaceColor', 'interp', 'EdgeColor', 'none')
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]');
title('Medium Stiff Platform');
title('$\omega_0 \approx \omega_0^\prime$');
view([0 0 1]);
set(gca,'ColorScale','log')
caxis([1e-3, 1]);
@@ -924,7 +921,7 @@ Let's fix $k = 10^7\ [N/m]$, $m = 100\ [kg]$ and see the evolution of $|w_{iI}(j
ax3 = subplot(1, 3, 3);
surf(freqs, xi, wiI_c_stiff', 'FaceColor', 'interp', 'EdgeColor', 'none')
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
title('Stiff Platform');
title('$\omega_0 \gg \omega_0^\prime$');
view([0 0 1]);
set(gca,'ColorScale','log')
caxis([1e-3 1e0]);
@@ -983,6 +980,7 @@ Let's fix $k = 10^7\ [N/m]$, $\xi = \frac{c}{2\sqrt{km}} = 0.1$ and see the evol
[[file:figs/inverse_multiplicative_uncertainty_norm_m.png]]
** Conclusion
#+begin_important
If the goal is to have an acceptable ($<10\%$) uncertainty on the plant until the highest frequency, two design choice for the isolation platform are possible:
- a very soft isolation platform $\omega_0 \ll \omega_0^\prime$
- a very stiff isolation platform $\omega_0 \gg \omega_0^\prime$
@@ -992,6 +990,7 @@ If a very soft isolation platform is used, the uncertainty due to the support's
If a very stiff isolation platform is used, the uncertainty will be high around $\omega_0^\prime$ and may reach unacceptable value.
It will then be high around $\omega_0$ and probably be higher than one.
Thus, if a stiff isolation platform is used, the recommendation is to have the largest possible resonance frequency, as the control bandwidth will be limited by the first resonance of the isolation platform (if not already limited by the resonance of the support).
#+end_important
* Numerical Analysis for the NASS :noexport:
<<sec:nass>>