Re-read the introduction and feedback section

index.org

@@ -34,7 +34,8 @@
 :END:
 
 * Introduction                                                        :ignore:
-The overall objective is to design a nano-hexapod an the associated control architecture that allows the stabilization of samples down to $\approx 10nm$ in presence of disturbances and system variability.
+In this document are gathered and summarized all the developments done for the design of the Nano Active Stabilization System.
+This consists of a nano-hexapod and an associated control architecture that are used to stabilize samples down to the nano-meter level in presence of disturbances.
 
 
 To understand the design challenges of such system, a short introduction to Feedback control is provided in Section [[sec:feedback_introduction]].
@@ -43,15 +44,14 @@ The mathematical tools (Power Spectral Density, Noise Budgeting, ...) that will
 
 To be able to develop both the nano-hexapod and the control architecture in an optimal way, we need a good estimation of:
 - the micro-station dynamics (Section [[sec:micro_station_dynamics]])
-- the frequency content of the important source of disturbances in play such as vibration of stages and ground motion (Section [[sec:identification_disturbances]])
+- the frequency content of the sources of disturbances such as vibrations induced by the micro-station's stages and ground motion (Section [[sec:identification_disturbances]])
 
 
-We then develop a model of the system that must represent all the important physical effects in play.
+A model of the micro-station is then developed and tuned using the previous estimations (Section [[sec:multi_body_model]]).
-Such model is presented in Section [[sec:multi_body_model]].
+The nano-hexapod is further included in the model.
 
 
-A modular model of the nano-hexapod is then included in the system.
+The effects of the nano-hexapod characteristics on the system dynamics are then studied.
-The effects of the nano-hexapod characteristics on the dynamics are then studied.
 Based on that, an optimal choice of the nano-hexapod stiffness is made (Section [[sec:nano_hexapod_design]]).
 
 
@@ -61,8 +61,9 @@ 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]]).
+** Introduction                                                      :ignore:
-This should highlight the challenges in terms of combined performance and robustness.
+In this section, some basics of *feedback systems* are first introduced (Section [[sec:feedback]]).
+This should highlight the challenges of the required combined performance and robustness.
 
 
 In Section [[sec:noise_budget]] is introduced the *dynamic error budgeting* which is a powerful tool that allows to derive the total error in a dynamic system from multiple disturbance sources.
@@ -72,14 +73,17 @@ This tool will be widely used throughout this study to both predict the performa
 <<sec:feedback>>
 
 *** Introduction                                                    :ignore:
+The use of Feedback control in a motion system required to use some sensors to monitor the actual status of the system and actuators to modifies this status.
+
 The use of feedback control as several advantages and pitfalls that are listed below (taken from cite:schmidt14_desig_high_perfor_mechat_revis_edition):
 
-- *Advantages*:
+*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
+    Disturbances inducing 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*:
+
+*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/
@@ -91,13 +95,13 @@ The use of feedback control as several advantages and pitfalls that are listed b
 
 *** Simplified Feedback Control Diagram for the NASS
 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 to be controlled)
-- $d$: the disturbances affecting $y$ (ground motion, vibration of stages)
+- $d$: the disturbances affecting $y$ (ground motion, stages' vibrations)
 - $n$: the noise of the sensor measuring $y$
 - $r$: the reference signal, corresponding to the wanted $y$
 - $\epsilon = r - y$: the position error
 
-And the dynamical blocks are:
+The /dynamical/ blocks are:
 - $G$: representing the dynamics from forces/torques applied by the nano-hexapod 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)
@@ -129,11 +133,11 @@ 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:
+Without the use of feedback (i.e. without the 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 $\approx 10nm$).
+which is, in the case of the NASS 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.
 
@@ -159,10 +163,8 @@ From Eq. eqref:eq:closed_loop_error representing the closed-loop system behavior
 - the measurement noise $n$ is injected and multiplied by a factor $T$
 
 Ideally, we would like to design the controller $K$ such that:
-- $|S|$ is small to limit the effect of disturbances
+- $|S|$ is small to *reduce the effect of disturbances*
-- $|T|$ is small to limit the injection of sensor noise
+- $|T|$ is small to *limit the injection of sensor noise*
-
-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:
@@ -176,14 +178,15 @@ There is therefore a *trade-off between the disturbance rejection and the measur
 
 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
+*At low frequency* (inside the control bandwidth):
-  - $|T| \approx 1$ and all the sensor noise is transmitted
+- $|S|$ can be made small and thus the effect of disturbances is reduced
-- *At high frequency* (outside the control bandwidth):
+- $|T| \approx 1$ and all the sensor noise is transmitted
-  - $|S| \approx 1$ and the feedback system does not reduce the effect of disturbances
+*At high frequency* (outside the control bandwidth):
-  - $|T|$ is small and thus the sensor noise is filtered
+- $|S| \approx 1$ and the feedback system does not reduce the effect of disturbances
-- *Near the crossover frequency* (between the two frequency bands):
+- $|T|$ is small and thus the sensor noise is filtered
-  - The effect of disturbances is increased
+*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}
@@ -228,16 +231,16 @@ We can observe that $|S|$ and $|T|$ exhibit different behaviors depending on the
 *** Trade off: Robustness / Performance
 <<sec:perf_robust_tradeoff>>
 
-As shown in the previous section, the effect of disturbances is reduced /inside/ the control bandwidth.
+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).
+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 usually large effects).
 
-The next important question is *what effects do limit the attainable control bandwidth?*
+The next important question is therefore *what limits 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*.
+The main issue it that for stability reasons, *the system dynamics 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]]).
+For mechanical systems, this generally means that the 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
@@ -250,30 +253,32 @@ For the NASS, the possible changes in the system are:
 - 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, ...)
 
-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.
+The nano-hexapod and the control architecture have to be developed in such a way that the feedback system remains stable and exhibit acceptable performance for all these possible changes in the system.
 
 This problem of *robustness* represent one of the main challenge for the design of the NASS.
 
-# 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.
-
 ** 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
+The dynamic error budgeting is a powerful tool to study the effects of multiple error sources (i.e. disturbances and measurement noise) and to predict how much these effects are reduced by a feedback system.
 
-To understand how to use and understand it, the Power Spectral Density and the Cumulative Power Spectrum are first introduced.
+The dynamic error budgeting uses two important mathematical functions: the *Power Spectral Density* and the *Cumulative Power Spectrum*.
-Then, is shown how does multiple error sources are combined and modified by dynamical systems.
 
-Finally,
+After these two functions are introduced (in Sections [[sec:psd]] and [[sec:cps]]), is shown how do multiple error sources are combined and modified by dynamical systems (in Section [[sec:psd_lti_system]] and [[sec:psd_combined_signals]]).
+
+Finally, the dynamic noise budgeting for the NASS is derived in Section [[sec:dynamic_noise_budget]].
 
 *** Power Spectral Density
+<<sec:psd>>
+
 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:
-\[ S_{xx}(\omega) = \int_{-\infty}^{\infty} R_{xx}(\tau) e^{-j \omega \tau} d\tau \ \frac{[\text{unit of } x]^2}{\text{Hz}} \]
+\[ S_{xx}(\omega) = \int_{-\infty}^{\infty} R_{xx}(\tau) e^{-j \omega \tau} d\tau \quad \frac{[\text{unit of } x]^2}{\text{Hz}} \]
 
 The PSD $S_{xx}(\omega)$ 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:
+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}(\omega) d\omega}
 \end{equation}
@@ -284,6 +289,8 @@ One can also integrate the infinitesimal power $S_{xx}(\omega)d\omega$ over a fi
 \end{equation}
 
 *** Cumulative Power Spectrum
+<<sec:cps>>
+
 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 of } x]^2
@@ -297,10 +304,13 @@ An alternative definition of the Cumulative Power Spectrum can be used where the
 \end{equation}
 And thus $CPS_x(f)$ represents the power in the signal $x$ for frequencies above $f$.
 
+The cumulative
 
-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.
+The Cumulative Power Spectrum is generally shown as a function of frequency, and is used to determine at which frequencies the effect of disturbances must be reduced, and thus the approximate required control bandwidth.
+
+*** Modification of a signal's PSD when going through a dynamical system
+<<sec:psd_lti_system>>
 
-*** 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]]).
 
 #+begin_src latex :file psd_lti_system.pdf
@@ -323,6 +333,8 @@ The Power Spectral Density of the output signal $y$ can be computed using:
 \end{equation}
 
 *** PSD of combined signals
+<<sec:psd_combined_signals>>
+
 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 $v$ (can be easily shown from the definition of the PSD):
@@ -345,6 +357,8 @@ We have that the PSD of $y$ is equal to sum of the PSD and $u$ and the PSD of $v
 [[file:figs/psd_sum.png]]
 
 *** Dynamic Noise Budgeting
+<<sec:dynamic_noise_budget>>
+
 Let's consider the Feedback architecture in Figure [[fig:classical_feedback_small]] where the position error $\epsilon$ is equal to:
 \[ \epsilon = S r + T n - G_d S d \]
 
@@ -360,9 +374,10 @@ And we can compute the RMS value of the residual motion using:
 
 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:
-  - $S_{dd}$: disturbances, this will be done in Section [[sec:identification_disturbances]]
+  - The disturbances $S_{dd}$: this will be done in Section [[sec:identification_disturbances]]
-  - $S_{nn}$: sensor noise, this can be estimated from the sensor data-sheet
+  - The sensor noise $S_{nn}$: this can be estimated from the sensor data-sheet
-  - $S_{rr}$: which is a deterministic signal that we choose. For simple tomography experiment, we can consider that it is equal to $0$
+  - The wanted sample's motion $S_{rr}$: this is a deterministic signal that we choose.
+    For a simple tomography experiment, we can consider that it is equal to $0$ as we only want to compensate all the sample's vibrations
 - 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]]
@@ -1232,6 +1247,11 @@ A more complete study of the control of the NASS is performed [[https://tdehaeze
 ** General Conclusion
 
 
+** Sensor Noise introduced by the Metrology
+Say that is will introduce noise inside the bandwidth (100Hz)
+This should not be significant.
+
+
 ** Further Work