Add some notes about feedback systems

index.org

@@ -37,7 +37,7 @@
 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.
 
 
-To understand the design challenges of such system, a short introduction to Feedback control is provided in Section [[sec:feedback]].
+To understand the design challenges of such system, a short introduction to Feedback control is provided in Section [[sec:feedback_introduction]].
 The mathematical tools (Power Spectral Density, Noise Budgeting, ...) that will be used throughout this study are also introduced.
 
 
@@ -58,11 +58,51 @@ Based on that, an optimal choice of the nano-hexapod stiffness is made (Section
 Finally, using the optimally designed nano-hexapod, a robust control architecture is developed.
 Simulations are performed to show that this design gives acceptable performance and the required robustness (Section [[sec:robust_control_architecture]]).
 
-* Feedback Systems and Noise budgeting
+* Introduction to Feedback Systems and Noise budgeting
+<<sec:feedback_introduction>>
+
+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.
+
+
+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.
+This tool will be widely used throughout this study to both predict the performances and identify the effects that do limit the performances.
+
+** Feedback System
 <<sec:feedback>>
 
-** Simple Feedback System
+*** Introduction                                                    :ignore:
-We usually analyze dynamical systems in the frequency domain using the Laplace transform.
+
+From cite:schmidt14_desig_high_perfor_mechat_revis_edition:
+
+Feedback control has the following 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
+
+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)
+- $d$ the disturbances affecting $y$ (ground motion, vibration of stages)
+- $n$ 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:
+- $G$ 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 $y$
+- $K$ representing the controller to be designed
 
 #+begin_src latex :file classical_feedback_small.pdf
   \begin{tikzpicture}
@@ -87,15 +127,12 @@ We usually analyze dynamical systems in the frequency domain using the Laplace t
 #+end_src
 
 #+name: fig:classical_feedback_small
-#+caption: Figure caption
+#+caption: Block Diagram of a simple feedback system
 #+RESULTS:
 [[file:figs/classical_feedback_small.png]]
 
-- $y$ is the relative position of the sample with respect to the granite
+*** How does the feedback loop is modifying the system behavior?
-- $d$ is the disturbances affecting $y$ (ground motion, vibration of stages)
+
-- $n$ is the noise of the sensor measuring $y$
-- $r$ is the reference signal, corresponding to the wanted $y$
-- we note $\epsilon = r - y$ the position error
 
 \[ \epsilon = \frac{1}{1 + GK} r + \frac{GK}{1 + GK} n - \frac{G_d}{1 + GK} d \]
 
@@ -106,6 +143,10 @@ We usually note:
 \end{align}
 
 $S$ is called the sensibility transfer function and $T$ the transmissibility transfer function.
+We can easily see that
+\[ S + T = 1 \]
+and thus, we cannot have $S$ and $T$ small at the same time.
+
 
 And we have:
 \[ \epsilon = S r + T n - G_d S d \]
@@ -117,6 +158,7 @@ However, when $|S|$ is small, $|T| \approx 1$ and all the sensor noise is transm
 #+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[] (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[]{$\sim \left| GK \right|^{-1}$};
@@ -125,10 +167,10 @@ 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,-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[dashed] (-0.5, -2.7) rectangle (5.5, 1.4);
     \end{scope}
 
     \begin{scope}[shift={(6.4, 0)}]
+      \draw[dashed, fill=white] (-0.5, -2.7) 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[]{$\sim 1$};
@@ -137,19 +179,47 @@ 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[dashed] (-0.5, -2.7) rectangle (5.5, 1.4);
     \end{scope}
   \end{tikzpicture}
 #+end_src
 
 #+name: fig:h-infinity-2-blocs-constrains
-#+caption: Figure caption
+#+caption: Typical shape 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 $G_d$.
 
-** Noise Budgeting
+*** 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.
+
+#+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:
+- bandwidth is limited at a frequency where the behavior of the system is not known
+
+Predictible system.
+
+For instance, ASML, everything is calibrated (wafer, some size, mass, etc...)
+
+Here, the main difficulty is that we want a very high performance system that is robust to change of:
+- Micro Station Configuration: position of the stages, change of on stage
+- Payload mass and dynamics
+- Spindle's rotation speed
+
+** Dynamic error budgeting
+<<sec:noise_budget>>
+
 *** Introduction                                                    :ignore:
 
 *** Power Spectral Density
@@ -253,36 +323,6 @@ To estimate the PSD of the position error $\epsilon$ and thus the RMS residual m
   - $S_{dd}$
 - The dynamics of the system $G$, $G_d$ and the controller $K$ (or alternatively $S$, $T$ and $G_d$)
 
-** Trade off Robustness / Performance
-If we want high level of performance, the experimental conditions should be carefully controlled.
-
-#+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:
-- bandwidth is limited at a frequency where the behavior of the system is not known
-
-Predictible system.
-
-For instance, ASML, everything is calibrated (wafer, some size, mass, etc...)
-
-Here, the main difficulty is that we want a very high performance system that is robust to change of:
-- Micro Station Configuration: position of the stages, change of on stage
-- Payload mass and dynamics
-- Spindle's rotation speed
-
-** 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.
-
-
-- [ ] Add a
-
-we will generally decrease the effect of the disturbances
-
-- [ ] Find the citation where it is said that the bandwidth is the consequence of the wanted disturbance rejection at some lower frequency
-
 * Identification of the Micro-Station Dynamics
 <<sec:micro_station_dynamics>>
 
@@ -551,3 +591,9 @@ https://tdehaeze.github.io/nass-simscape/optimal_stiffness_control.html
   - [ ] A zoom on at the nano-meter level to see how the wanted position is moving
 
 ** Conclusion
+* Further notes
+Soft granite
+
+nano-focusing lenses
+
+Detector