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Re-read the introduction and feedback section

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@ -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.
Such model is presented in Section [[sec:multi_body_model]].
A model of the micro-station is then developed and tuned using the previous estimations (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 dynamics are then studied.
The effects of the nano-hexapod characteristics on the system 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]]).
This should highlight the challenges in terms of combined performance and robustness.
** Introduction :ignore:
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)
- $d$: the disturbances affecting $y$ (ground motion, vibration of stages)
- $y$: the relative position of the sample with respect to the granite (the quantity to be controlled)
- $d$: 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
- $|T|$ 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.
- $|S|$ is small to *reduce the effect of disturbances*
- $|T|$ is small to *limit the injection of sensor noise*
*** Trade off: Disturbance Reduction / Noise Injection
We have from the definition of $S$ and $T$ that:
@ -176,13 +178,14 @@ 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):
*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):
*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):
*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
@ -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.
Then, is shown how does multiple error sources are combined and modified by dynamical systems.
The dynamic error budgeting uses two important mathematical functions: the *Power Spectral Density* and the *Cumulative Power Spectrum*.
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]]
- $S_{nn}$: sensor noise, 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 disturbances $S_{dd}$: this will be done in Section [[sec:identification_disturbances]]
- The sensor noise $S_{nn}$: this can be estimated from the sensor data-sheet
- 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