Finish explanation of noise budgeting

index.org

@@ -61,7 +61,7 @@ Simulations are performed to show that this design gives acceptable performance
 * Introduction to Feedback Systems and Noise budgeting
 <<sec:feedback_introduction>>
 
-In this section, we first introduce some basics of feedback systems (Section [[sec:feedback]]).
+In this section, we first introduce some basics of *feedback systems* (Section [[sec:feedback]]).
 This should highlight the challenges in terms of combined performance and robustness.
 
 
@@ -72,37 +72,35 @@ This tool will be widely used throughout this study to both predict the performa
 <<sec:feedback>>
 
 *** Introduction                                                    :ignore:
+The use of feedback control as several advantages and pitfalls that are listed below (taken from cite:schmidt14_desig_high_perfor_mechat_revis_edition):
 
-From cite:schmidt14_desig_high_perfor_mechat_revis_edition:
+- *Advantages*:
+  - *Reduction of the effect of disturbances*:
+    Disturbances affecting the sample vibrations are observed by the sensor signal, and therefore the feedback controller can compensate for them
+  - *Handling of uncertainties*:
+    Feedback controlled systems can also be designed for /robustness/, which means that the stability and performance requirements are guaranteed even for parameter variation of the controller mechatronics system
+- *Pitfalls*:
+  - *Limited reaction speed*:
+    A feedback controller reacts on the difference between the reference signal (wanted motion) and the measurement (actual motion), which means that the error has to occur first /before/ the controller can correct for it.
+    The limited reaction speed means that the controller will be able to compensate the positioning errors only in some frequency band, called the controller /bandwidth/
+  - *Feedback of noise*:
+    By closing the loop, the sensor noise is also fed back and will induce positioning errors
+  - *Can introduce instability*:
+    Feedback control can destabilize a stable plant.
+    Thus the /robustness/ properties of the feedback system must be carefully guaranteed
 
-Feedback control has the following advantages:
+*** Simplified Feedback Control Diagram for the NASS
-- *Reduction of the effect of disturbances*:
-  Disturbances affecting the sample vibrations are observed by the sensor signal, and therefore the feedback controller can compensate for them
-- *Handling of uncertainties*:
-  Feedback controlled systems can also be designed for /robustness/, which means that the stability and performance requirements are guaranteed even for parameter variation of the controller mechatronics system
-
-But it also has some pitfalls:
-- *Limited reaction speed*:
-  A feedback controller reacts on the difference between the reference signal (wanted motion) and the measurement (actual motion), which means that the error has to occur first before the controller can correct for it.
-  The limited reaction speed means that the controller will be able to compensate the positioning errors only in some frequency band, called the *controller bandwidth*
-- *Feedback of noise*:
-  By closing the loop, the sensor noise is also fed back and will introduce positioning errors
-- *Can introduce instability*:
-  Feedback control can destabilize a stable plant.
-  Thus the /robustness/ properties of the feedback system must be carefully guaranteed
-
-*** Introduction to Feedback Control
 Let's consider the block diagram shown in Figure [[fig:classical_feedback_small]] where the signals are:
-- $y$ the relative position of the sample with respect to the granite (the quantity we wish to control)
+- $y$: the relative position of the sample with respect to the granite (the quantity we wish to control)
-- $d$ the disturbances affecting $y$ (ground motion, vibration of stages)
+- $d$: the disturbances affecting $y$ (ground motion, vibration of stages)
-- $n$ the noise of the sensor measuring $y$
+- $n$: the noise of the sensor measuring $y$
-- $r$ the reference signal, corresponding to the wanted $y$
+- $r$: the reference signal, corresponding to the wanted $y$
-- $\epsilon = r - y$ the position error
+- $\epsilon = r - y$: the position error
 
 And the dynamical blocks are:
-- $G$ representing the dynamics from forces/torques applied by the nano-hexapod to the relative position sample/granite $y$
+- $G$: representing the dynamics from forces/torques applied by the nano-hexapod to the relative position sample/granite $y$
-- $G_d$ representing the dynamics from the disturbances (e.g. ground motion) to the relative position sample/granite $y$
+- $G_d$: representing how the disturbances (e.g. ground motion) are affecting the relative position sample/granite $y$
-- $K$ representing the controller to be designed
+- $K$: representing the controller (to be designed)
 
 #+begin_src latex :file classical_feedback_small.pdf
   \begin{tikzpicture}
@@ -131,9 +129,16 @@ And the dynamical blocks are:
 #+RESULTS:
 [[file:figs/classical_feedback_small.png]]
 
+Without the use of feedback (i.e. nano-hexapod), the disturbances will induce a sample motion error equal to:
+\begin{equation}
+  y = G_d d \label{eq:open_loop_error}
+\end{equation}
+which is out of the specifications (micro-meter range compare to the required $\approx 10nm$).
+
+In the next section, we see how the use of the feedback system permits to lower the effect of the disturbances $d$ on the sample motion error.
+
 *** How does the feedback loop is modifying the system behavior?
-
+If we write down the position error signal $\epsilon = r - y$ as a function of the reference signal $r$, the disturbances $d$ and the measurement noise $n$ (using the feedback diagram in Figure [[fig:classical_feedback_small]]), we obtain:
-
 \[ \epsilon = \frac{1}{1 + GK} r + \frac{GK}{1 + GK} n - \frac{G_d}{1 + GK} d \]
 
 We usually note:
@@ -141,24 +146,49 @@ We usually note:
   S &= \frac{1}{1 + GK} \\
   T &= \frac{GK}{1 + GK}
 \end{align}
+where $S$ is called the sensibility transfer function and $T$ the transmissibility transfer function.
 
-$S$ is called the sensibility transfer function and $T$ the transmissibility transfer function.
+And the position error can be rewritten as:
-We can easily see that
+\begin{equation}
-\[ S + T = 1 \]
+  \epsilon = S r + T n - G_d S d \label{eq:closed_loop_error}
-and thus, we cannot have $S$ and $T$ small at the same time.
+\end{equation}
 
 
-And we have:
+From Eq. eqref:eq:closed_loop_error representing the closed-loop system behavior, we can see that:
-\[ \epsilon = S r + T n - G_d S d \]
+- the effect of disturbances $d$ on $\epsilon$ is multiplied by a factor $S$ compared to the open-loop case
+- the measurement noise $n$ is injected and multiplied by a factor $T$
 
-Thus, we usually want $|S|$ small such that the effect of disturbances are reduced down to acceptable levels and such that the system is able to follow the change of reference with only small tracking errors.
+Ideally, we would like to design the controller $K$ such that:
+- $|S|$ is small to limit the effect of disturbances
+- $|T|$ is small to limit the injection of sensor noise
 
-However, when $|S|$ is small, $|T| \approx 1$ and all the sensor noise is transmitted to the position error.
+As shown in the next section, there is a trade-off between the disturbance reduction and the noise injection.
+
+*** Trade off: Disturbance Reduction / Noise Injection
+We have from the definition of $S$ and $T$ that:
+\begin{equation}
+  S + T = \frac{1}{1 + GK} + \frac{GK}{1 + GK} = 1
+\end{equation}
+meaning that we cannot have $|S|$ and $|T|$ small at the same time.
+
+There is therefore a *trade-off between the disturbance rejection and the measurement noise filtering*.
+
+
+Typical shapes of $|S|$ and $|T|$ as a function of frequency are shown in Figure [[fig:h-infinity-2-blocs-constrains]].
+We can observe that $|S|$ and $|T|$ exhibit different behaviors depending on the frequency band:
+- *At low frequency* (inside the control bandwidth):
+  - $|S|$ can be made small and thus the effect of disturbances is reduced
+  - $|T| \approx 1$ and all the sensor noise is transmitted
+- *At high frequency* (outside the control bandwidth):
+  - $|S| \approx 1$ and the feedback system does not reduce the effect of disturbances
+  - $|T|$ is small and thus the sensor noise is filtered
+- *Near the crossover frequency* (between the two frequency bands):
+  - The effect of disturbances is increased
 
 #+begin_src latex :file h-infinity-2-blocs-constrains.pdf
   \begin{tikzpicture}
     \begin{scope}[shift={(0, 0)}]
-      \draw[dashed, fill=white] (-0.5, -2.7) rectangle (5.5, 1.4);
+      \draw[dashed, fill=white] (-0.5, -3.4) rectangle (5.5, 1.4);
       \draw[] (2.5, 1.0) node[]{$\left| S(j\omega) \right|$};
       \draw[fill=blue!20] (-0.2, -2.5) rectangle (1.4, 0.5);
       \draw[] (0.6, -0.5) node[]{$\sim \left| GK \right|^{-1}$};
@@ -166,11 +196,15 @@ However, when $|S|$ is small, $|T| \approx 1$ and all the sensor noise is transm
       \draw[] (4.5, -0.5) node[]{$\sim 1$};
       \draw[fill=red!20] (2.5, 0.15) circle (0.15);
       \draw[dashed] (-0.4, 0) -- (5.4, 0);
-      \draw [] (0,-2) to[out=45,in=180+45] (2,0) to[out=45,in=180] (2.5,0.3) to[out=0,in=180] (3.5,0) to[out=0,in=180] (5, 0);
+      \draw [] (0,-2) to[out=45,in=180+45] (2,0) to[out=45,in=180] (2.5,0.3)
+      to[out=0,in=180] (3.5,0) to[out=0,in=180] (5, 0);
+      \draw[<->] (-0.2, -2.8) -- node[midway, below, align=center]{\footnotesize Low Freq. } (1.8, -2.8);
+      \draw[<->] (1.8, -2.8) --  node[midway, below, align=center]{\footnotesize Cross Over} (3.2, -2.8);
+      \draw[<->] (3.2, -2.8) --  node[midway, below, align=center]{\footnotesize High Freq.} (5.2, -2.8);
     \end{scope}
 
     \begin{scope}[shift={(6.4, 0)}]
-      \draw[dashed, fill=white] (-0.5, -2.7) rectangle (5.5, 1.4);
+      \draw[dashed, fill=white] (-0.5, -3.4) rectangle (5.5, 1.4);
       \draw[] (2.5, 1.0) node[]{$\left| T(j\omega) \right|$};
       \draw[fill=red!20] (-0.2, -2.5) rectangle (1.4, 0.5);
       \draw[] (0.6, -0.5) node[]{$\sim 1$};
@@ -179,85 +213,92 @@ However, when $|S|$ is small, $|T| \approx 1$ and all the sensor noise is transm
       \draw[fill=red!20] (2.5, 0.15) circle (0.15);
       \draw[dashed] (-0.4, 0) -- (5.4, 0);
       \draw [] (0,0) to[out=0,in=180] (1.5,0) to[out=0,in=180] (2.5,0.3) to[out=0,in=-45] (3,0) to[out=-45,in=180-45] (5, -2);
+      \draw[<->] (-0.2, -2.8) -- node[midway, below, align=center]{\footnotesize Low Freq. } (1.8, -2.8);
+      \draw[<->] (1.8, -2.8) --  node[midway, below, align=center]{\footnotesize Cross Over} (3.2, -2.8);
+      \draw[<->] (3.2, -2.8) --  node[midway, below, align=center]{\footnotesize High Freq.} (5.2, -2.8);
     \end{scope}
   \end{tikzpicture}
 #+end_src
 
 #+name: fig:h-infinity-2-blocs-constrains
-#+caption: Typical shape and constrain of the Sensibility and Transmibility closed-loop transfer functions
+#+caption: Typical shapes and constrain of the Sensibility and Transmibility closed-loop transfer functions
 #+RESULTS:
 [[file:figs/h-infinity-2-blocs-constrains.png]]
 
-The nano-hexapod characteristics will change both $G$ and $G_d$.
+*** Trade off: Robustness / Performance
-
-*** Sensibility Transfer Function and Control Bandwidth
-When applying feedback in a system, it is much more convenient to look at things in the frequency domain.
-
-We will generally decrease the effect of the disturbances
-
-The bandwidth is the consequence of the wanted disturbance rejection at some lower frequency
-
-*** Trade off Robustness / Performance
 <<sec:perf_robust_tradeoff>>
-If we want high level of performance, the experimental conditions should be carefully controlled.
+
+As shown in the previous section, the effect of disturbances is reduced /inside/ the control bandwidth.
+
+Moreover, the slope of $|S(j\omega)|$ is limited for stability reasons (not explained here), and therefore a large control bandwidth is required to obtain sufficient disturbance rejection at lower frequencies (where the disturbances have large effects).
+
+The next important question is *what effects do limit the attainable control bandwidth?*
+
+
+The main issue it that for stability reasons, *the behavior of the mechanical system must be known with only small uncertainty in the vicinity of the crossover frequency*.
+
+For mechanical systems, this generally means that control bandwidth should take place before any appearing of flexible dynamics (Right part of Figure [[fig:oomen18_next_gen_loop_gain]]).
 
 #+name: fig:oomen18_next_gen_loop_gain
 #+caption: Envisaged developments in motion systems. In traditional motion systems, the control bandwidth takes place in the rigid-body region. In the next generation systemes, flexible dynamics are foreseen to occur within the control bandwidth. cite:oomen18_advan_motion_contr_precis_mechat
 [[file:figs/oomen18_next_gen_loop_gain.png]]
 
-Limitation of feedback control:
+This also means that *any possible change in the system should have a small impact on the system dynamics in the vicinity of the crossover*.
-- bandwidth is limited at a frequency where the behavior of the system is not known
 
-Predictible system.
+For the NASS, the possible changes in the system are:
+- a modification of the payload mass and dynamics
+- a change of experimental condition: spindle's rotation speed, position of each micro-station's stage
+- a change in the micro-station dynamics (change of mechanical elements, aging effect, ...)
 
-For instance, ASML, everything is calibrated (wafer, some size, mass, etc...)
+The nano-hexapod and the control architecture have to be developed such that the feedback system remains stable and exhibit acceptable performance for all these possible changes in the system.
 
-Here, the main difficulty is that we want a very high performance system that is robust to change of:
+This problem of *robustness* represent one of the main challenge for the design of the NASS.
-- Micro Station Configuration: position of the stages, change of on stage
+
-- Payload mass and dynamics
+# High performance mechatronics systems (e.g. Wafer stages, or Atomic Force Microscopes) are usually developed in such a way that their mechanical behavior is extremely well known up to high frequency and such that the experimental conditions are usually be carefully controlled.
-- Spindle's rotation speed
 
 ** Dynamic error budgeting
 <<sec:noise_budget>>
 
 *** Introduction                                                    :ignore:
+The dynamic error budgeting is a powerful tool to study the effect of multiple error sources and to see how the feedback system does reduce the effect
+
+To understand how to use and understand it, the Power Spectral Density and the Cumulative Power Spectrum are first introduced.
+Then, is shown how does multiple error sources are combined and modified by dynamical systems.
+
+Finally,
 
 *** Power Spectral Density
-The *Power Spectral Density* (PSD) $S_{xx}(f)$ of the time domain $x(t)$ (in $[m]$) can be computed using the following equation:
+The *Power Spectral Density* (PSD) $S_{xx}(f)$ of the time domain signal $x(t)$ is defined as the Fourier transform of the autocorrelation function:
-\[ S_{xx}(f) = \frac{1}{f_s} \sum_{m=-\infty}^{\infty} R_{xx}(m) e^{-j 2 \pi m f / f_s} \ \left[\frac{m^2}{\text{Hz}}\right] \]
+\[ S_{xx}(\omega) = \int_{-\infty}^{\infty} R_{xx}(\tau) e^{-j \omega \tau} d\tau \ \frac{[\text{unit of } x]^2}{\text{Hz}} \]
-where
-- $f_s$ is the sampling frequency in $[Hz]$
-- $R_{xx}$ is the autocorrelation
 
-
+The PSD $S_{xx}(\omega)$ represents the *distribution of the (average) signal power over frequency*.
-The PSD $S_{xx}(f)$ represents the distribution of the (average) signal power over frequency.
 
 Thus, the total power in the signal can be obtained by integrating these infinitesimal contributions, the Root Mean Square (RMS) value of the signal $x(t)$ is then:
 \begin{equation}
-  x_{\text{rms}} = \sqrt{\int_{0}^{\infty} S_{xx}(f) df} \ [m,\text{rms}]
+  x_{\text{rms}} = \sqrt{\int_{0}^{\infty} S_{xx}(\omega) d\omega}
 \end{equation}
 
-One can also integrate the infinitesimal power $S_{xx}(f)df$ over a finite frequency band to obtain the power of the signal $x$ in that frequency band:
+One can also integrate the infinitesimal power $S_{xx}(\omega)d\omega$ over a finite frequency band to obtain the power of the signal $x$ in that frequency band:
 \begin{equation}
-  P_{f_1,f_2} = \int_{f_1}^{f_2} S_{xx}(f) df \quad [m^2]
+  P_{f_1,f_2} = \int_{f_1}^{f_2} S_{xx}(\omega) d\omega \quad [\text{unit of } x]^2
 \end{equation}
 
 *** Cumulative Power Spectrum
 The *Cumulative Power Spectrum* is the cumulative integral of the Power Spectral Density starting from $0\ \text{Hz}$ with increasing frequency:
 \begin{equation}
-  CPS_x(f) = \int_0^f S_{xx}(\nu) d\nu \quad [\text{unit}^2]
+  CPS_x(f) = \int_0^f S_{xx}(\nu) d\nu \quad [\text{unit of } x]^2
 \end{equation}
 The Cumulative Power Spectrum taken at frequency $f$ thus represent the power in the signal in the frequency band $0$ to $f$.
 
 
 An alternative definition of the Cumulative Power Spectrum can be used where the PSD is integrated from $f$ to $\infty$:
 \begin{equation}
-  CPS_x(f) = \int_f^\infty S_{xx}(\nu) d\nu \quad [\text{unit}^2]
+  CPS_x(f) = \int_f^\infty S_{xx}(\nu) d\nu \quad [\text{unit of } x]^2
 \end{equation}
 And thus $CPS_x(f)$ represents the power in the signal $x$ for frequencies above $f$.
 
 
-The Cumulative Power Spectrum can be used to determine in which frequency band the effect of disturbances should be reduced and the approximated required control bandwidth in order to obtained some specified vibration amplitude.
+The Cumulative Power Spectrum will be used to determine in which frequency band the effect of disturbances should be reduced, and thus the approximate required control bandwidth.
 
 *** Modification of a signal's PSD when going through an LTI system
 Let's consider a signal $u$ with a PSD $S_{uu}$ going through a LTI system $G(s)$ that outputs a signal $y$ with a PSD (Figure [[fig:psd_lti_system]]).
@@ -272,7 +313,7 @@ Let's consider a signal $u$ with a PSD $S_{uu}$ going through a LTI system $G(s)
 #+end_src
 
 #+NAME: fig:psd_lti_system
-#+CAPTION:
+#+CAPTION: LTI dynamical system $G(s)$ with input signal $u$ and output signal $y$
 #+RESULTS:
 [[file:figs/psd_lti_system.png]]
 
@@ -282,9 +323,9 @@ The Power Spectral Density of the output signal $y$ can be computed using:
 \end{equation}
 
 *** PSD of combined signals
-Let's consider a signal $y$ that is the sum of two *uncorrelated* signals $u$ and $v$.
+Let's consider a signal $y$ that is the sum of two *uncorrelated* signals $u$ and $v$ (Figure [[fig:psd_sum]]).
 
-We have that the PSD of $y$ is equal to sum of the PSD and $u$ and the PSD of $v$:
+We have that the PSD of $y$ is equal to sum of the PSD and $u$ and the PSD of $v$ (can be easily shown from the definition of the PSD):
 \[ S_{yy} = S_{uu} + S_{vv} \]
 
 #+begin_src latex :file psd_sum.pdf
@@ -298,30 +339,33 @@ We have that the PSD of $y$ is equal to sum of the PSD and $u$ and the PSD of $v
   \end{tikzpicture}
 #+end_src
 
+#+name: fig:psd_sum
+#+caption: $y$ as the sum of two signals $u$ and $v$
 #+RESULTS:
 [[file:figs/psd_sum.png]]
 
 *** Dynamic Noise Budgeting
-Let's consider the Feedback architecture,
+Let's consider the Feedback architecture in Figure [[fig:classical_feedback_small]] where the position error $\epsilon$ is equal to:
-
-The position error $\epsilon$ is equal to:
 \[ \epsilon = S r + T n - G_d S d \]
 
-If we suppose that the signals $r$, $n$ and $d$ are *uncorrelated*, the PSD of $\epsilon$ is:
+If we suppose that the signals $r$, $n$ and $d$ are *uncorrelated* (which is a good approximation in our case), the PSD of $\epsilon$ is:
 \[ S_{\epsilon \epsilon}(\omega) = |S(j\omega)|^2 S_{rr}(\omega) + |T(j\omega)|^2 S_{nn}(\omega) + |G_d(j\omega) S(j\omega)|^2 S_{dd}(\omega) \]
 
-And the RMS residual motion is equal to:
+And we can compute the RMS value of the residual motion using:
 \begin{align*}
   \epsilon_\text{rms} &= \sqrt{ \int_0^\infty S_{\epsilon\epsilon}(\omega) d\omega} \\
-  &= \sqrt{ \int_0^\infty |S(j\omega)|^2 S_{rr}(\omega) + |T(j\omega)|^2 S_{nn}(\omega) + |G_d(j\omega) S(j\omega)|^2 S_{dd}(\omega) d\omega }
+  &= \sqrt{ \int_0^\infty \Big( |S(j\omega)|^2 S_{rr}(\omega) + |T(j\omega)|^2 S_{nn}(\omega) + |G_d(j\omega) S(j\omega)|^2 S_{dd}(\omega) \Big) d\omega }
 \end{align*}
 
-To estimate the PSD of the position error $\epsilon$ and thus the RMS residual motion, we need:
+
+To estimate the PSD of the position error $\epsilon$ and thus the RMS residual motion (in closed-loop), we need to determine:
 - The Power Spectral Densities of the signals affecting the system:
-  - $S_{rr}$
+  - $S_{dd}$: disturbances, this will be done in Section [[sec:identification_disturbances]]
-  - $S_{nn}$
+  - $S_{nn}$: sensor noise, this can be estimated from the sensor data-sheet
-  - $S_{dd}$
+  - $S_{rr}$: which is a deterministic signal that we choose. For simple tomography experiment, we can consider that it is equal to $0$
-- The dynamics of the system $G$, $G_d$ and the controller $K$ (or alternatively $S$, $T$ and $G_d$)
+- The dynamics of the complete system comprising the micro-station and the nano-hexapod: $G$, $G_d$.
+  To do so, we need to identify the dynamics of the micro-station (Section [[sec:micro_station_dynamics]]), include this dynamics in a model (Section [[sec:multi_body_model]]) and add a model of the nano-hexapod to the model (Section [[sec:nano_hexapod_design]])
+- The controller $K$ that will be designed in Section [[sec:robust_control_architecture]]
 
 * Identification of the Micro-Station Dynamics
 <<sec:micro_station_dynamics>>
@@ -594,6 +638,10 @@ https://tdehaeze.github.io/nass-simscape/optimal_stiffness_control.html
 * Further notes
 Soft granite
 
-nano-focusing lenses
+Sensible to detector motion?
 
-Detector
+Common metrology frame for the nano-focusing optics and the measurement of the sample position?
+
+Cable forces?
+
+Slip-Ring noise?