2.6 Conclusion
+2.6 Conclusion
-3.4 Conclusion
+3.4 Conclusion
@@ -900,8 +900,8 @@ xlim([1, 500]); ylim([1e-10 1e
-
@@ -919,6 +919,9 @@ From Figure 20, we can see that a soft nano-hexapod \(
+From the total open-loop power spectral density of the sample’s motion error, we can estimate what is the required control bandwidth for the sample’s motion error to be reduced down to \(10nm\).
+
+The transfer functions corresponding to \(G_d\) are those identified in Section 2.
+
-As a first approximation, we can consider that the controller is designed in such a way that the loop gain \(KG\) is a pure integrator:
+As a first approximation, we can consider that the controller \(K\) is designed in such a way that the loop gain \(KG\) is a pure integrator:
\[ L_1(s) = K_1(s) G(s) = \frac{\omega_c}{s} \]
where \(\omega_c\) is the crossover frequency.
+In the next section, we see how the power spectral density of \(y\) is reduced as a function of the control bandwidth \(\omega_c\).
+This will help to determine what is the approximate control bandwidth required such that the rms value of \(y\) is below \(10nm\).
+
+Let’s first see how does the Cumulative Amplitude Spectrum of the sample’s motion error is modified by the control.
+
+In Figure 22 is shown the Cumulative Amplitude Spectrum of the sample’s motion error in Open-Loop and in Closed Loop for several control bandwidth (from 1Hz to 200Hz) and 4 different nano-hexapod stiffnesses.
+The controller used in this simulation is \(K_1\). The loop gain is then a pure integrator.
+
+In Figure 23 is shown the expected RMS value of the sample’s position error as a function of the control bandwidth, both for \(K_1\) (left plot) and \(K_2\) (right plot).
+As expected, it is shown that \(K_2\) performs better than \(K_1\).
+This Figure tells us how much control bandwidth is required to attain a certain level of performance, and that for all the considered nano-hexapod stiffnesses.
+
+The obtained required bandwidth (approximate upper and lower bounds) to obtained a sample’s motion error less than 10nm rms are gathered in Table 1.
+ Figure 22: Cumulative Amplitude Spectrum of the sample’s motion error in Open-Loop and in Closed Loop for several control bandwidth and 4 different nano-hexapod stiffnesses (png, pdf) Figure 23: Expected RMS value of the sample’s motion error \(E_z\) as a function of the control bandwidth when using \(K_1\) and \(K_2\) (png, pdf)
+From Figure 23 and Table 1, we can clearly see three different results depending on the nano-hexapod stiffness:
+ Created: 2020-04-07 mar. 19:33 Created: 2020-04-08 mer. 12:124.4 Conclusion
+4.4 Conclusion
5.1 Approximation of the effect of feedback on the motion error
@@ -940,10 +943,13 @@ The reduction of the impact of \(d\) on \(y\) thanks to feedback is described by
\begin{equation}
\frac{y}{d} = \frac{G_d}{1 + KG}
\end{equation}
+5.2 Reduction thanks to feedback - Required bandwidth
wc = 1*2*pi; % [rad/s]
-xic = 0.5;
+
+
+wc = [1, 5, 10, 20, 50, 100, 200];
-S1 = {zeros(length(wc), 1)};
-S2 = {zeros(length(wc), 1)};
-
-for j = 1:length(wc)
- L = (2*pi*wc(j))/s; % Simple integrator
- S1{j} = 1/(1 + L);
- L = ((2*pi*wc(j))/s)^2*(1 + s/(2*pi*wc(j)/2))/(1 + s/(2*pi*wc(j)*2));
- S2{j} = 1/(1 + L);
-end
-
+
+freqs = dist_f.f;
+
+
wc = logspace(0, 3, 100);
+
-freqs = dist_f.f;
+
+
+
+
+
+Nano-Hexapod stiffness [N/m]
+103
+104
+105
+106
+107
+108
+109
+
+
+
+Required wc with L1 [Hz]
+152
+305
+1000
+870
+933
+870
+870
+
+
+
+Required wc with L2 [Hz]
+57
+66
+152
+152
+248
+266
+248
+6 Conclusion
+
+
+
6 Conclusion
