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Add figure to show the opt gain on the root locus

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Thomas Dehaeze
2020-06-24 15:37:01 +02:00
parent db90a1b00a
commit 6dbad502bd
5 changed files with 1814 additions and 70 deletions
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@@ -240,11 +240,10 @@ As the rotation speed increases, one of the two resonant frequency goes to lower
| <<fig:campbell_diagram_real>> Real Part | <<fig:campbell_diagram_imag>> Imaginary Part |
#+name: fig:campbell_diagram
#+caption: Campbell Diagram
#+attr_latex: :scale 1
[[file:figs/campbell_diagram.pdf]]
# #+name: fig:campbell_diagram
# #+caption: Campbell Diagram
# #+attr_latex: :scale 1
# [[file:figs/campbell_diagram.pdf]]
# Bode Plots for different ratio wr/w0
@@ -341,10 +340,12 @@ This means that at low frequency, the system is decoupled (the force sensor remo
* Integral Force Feedback with High Pass Filters
** Modification of the Control Low
# Reference to Preumont where its done
# Equation with the new control law
# Explain why it is usually done and why it is done here: the problem is the high gain at low frequency => high pass filter
@@ -441,32 +442,58 @@ This means that at low frequency, the system is decoupled (the force sensor remo
# Large Stiffness decreases the attainable damping
# kp = 2mOmega to 5mOmega is ok
#+name: fig:root_locus_iff_kps
#+caption: Figure caption
#+attr_latex: :scale 1
[[file:figs/root_locus_iff_kps.pdf]]
# Example with kp = 5 m Omega
#+name: fig:root_locus_opt_gain_iff_kp
#+caption: Figure caption
#+attr_latex: :scale 1
[[file:figs/root_locus_opt_gain_iff_kp.pdf]]
* Direct Velocity Feedback
** Control Schematic
# Basic Idea of DVF
# Equation with the control law
#+name: fig:system_dvf
#+caption: Figure caption
#+attr_latex: :scale 1
[[file:figs/system_dvf.pdf]]
# Equivalent System is the same as Figure 1 (as increasing "c")
** Equations
# Write the equations
# Show that the rotation have somehow less impact on the plant than for IFF
# Eventually add a bode plot to show the effect of the rotation speed
** Relative Direct Velocity Feedback
# Unconditionally stable
# Arbitrary Damping can be added to the poles
#+name: fig:root_locus_dvf
#+caption: Figure caption
#+attr_latex: :scale 1
[[file:figs/root_locus_dvf.pdf]]
* Comparison of the Proposed Active Damping Techniques for Rotating Positioning Stages
**
**
@@ -475,6 +502,10 @@ This means that at low frequency, the system is decoupled (the force sensor remo
#+attr_latex: :scale 1
[[file:figs/comp_root_locus.pdf]]
** Transmissibility and Compliance
#+name: fig:comp_active_damping
#+caption: Comparison of the three proposed Active Damping Techniques
#+attr_latex: :environment subfigure :width 0.45\linewidth :align c

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paper/paper.pdf
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164
paper/paper.tex
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@@ -1,4 +1,4 @@
% Created 2020-06-23 mar. 19:34
% Created 2020-06-24 mer. 15:36
% Intended LaTeX compiler: pdflatex
\documentclass{ISMA_USD2020}
\usepackage[utf8]{inputenc}
@@ -23,6 +23,7 @@
\usepackage{amsmath,amssymb,amsfonts, cases}
\usepackage{algorithmic, graphicx, textcomp}
\usepackage{xcolor, import, hyperref}
\usepackage{subcaption}
\usepackage[USenglish, english]{babel}
\setcounter{footnote}{1}
\input{config.tex}
@@ -52,18 +53,17 @@
}
\section{Introduction}
\label{sec:org977317c}
\label{sec:orgbec19fa}
\label{sec:introduction}
Controller Poles are shown by black crosses (
\begin{tikzpicture} \node[cross out, draw=black, minimum size=1ex, line width=2pt, inner sep=0pt, outer sep=0pt] at (0, 0){}; \end{tikzpicture}
).
\cite{dehaeze18_sampl_stabil_for_tomog_exper}
\section{System Under Study}
\label{sec:org042e800}
\subsection{Rotating Positioning Platform}
\label{sec:org489e4b9}
Consider the rotating X-Y stage of Figure \ref{fig:rotating_xy_platform}.
\section{Dynamics of Rotating Positioning Platforms}
\label{sec:org81be86a}
\subsection{Studied Rotating Positioning Platform}
\label{sec:orgf8fad9b}
Consider the rotating X-Y stage of Figure \ref{fig:system}.
\begin{itemize}
\item \(k\): Actuator's Stiffness [N/m]
@@ -76,18 +76,11 @@ Consider the rotating X-Y stage of Figure \ref{fig:rotating_xy_platform}.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/system.pdf}
\caption{\label{fig:rotating_xy_platform}Figure caption}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.5\linewidth]{figs/cedrat_xy25xs.jpg}
\caption{\label{fig:cedrat_xy25xs}Figure caption}
\caption{\label{fig:system}Figure caption}
\end{figure}
\subsection{Equation of Motion}
\label{sec:orgb1836d5}
\label{sec:org926ba54}
The system has two degrees of freedom and is thus fully described by the generalized coordinates \(u\) and \(v\).
Let's express the kinetic energy \(T\) and the potential energy \(V\) of the mass \(m\) (neglecting the rotational energy):
@@ -132,7 +125,7 @@ Thus, the term \(- m\dot{\theta}^2\) acts like a negative stiffness (due to \tex
\subsection{Transfer Functions in the Laplace domain}
\label{sec:orgb1002ed}
\label{sec:org298e237}
\begin{subequations}
\begin{align}
@@ -171,7 +164,7 @@ With:
\subsection{Constant Rotating Speed}
\label{sec:orga4faf60}
\label{sec:org8d2eda6}
To simplify, let's consider a constant rotating speed \(\dot{\theta} = \Omega\) and thus \(\ddot{\theta} = 0\).
\begin{equation}
@@ -212,23 +205,38 @@ When the rotation speed in not null, the resonance frequency is duplicated into
As the rotation speed increases, one of the two resonant frequency goes to lower frequencies as the other one goes to higher frequencies (Figure \ref{fig:campbell_diagram}).
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/campbell_diagram.pdf}
\begin{subfigure}[c]{0.4\linewidth}
\includegraphics[width=\linewidth]{figs/campbell_diagram_real.pdf}
\caption{\label{fig:campbell_diagram_real} Real Part}
\end{subfigure}
\begin{subfigure}[c]{0.4\linewidth}
\includegraphics[width=\linewidth]{figs/campbell_diagram_imag.pdf}
\caption{\label{fig:campbell_diagram_imag} Imaginary Part}
\end{subfigure}
\caption{\label{fig:campbell_diagram}Campbell Diagram}
\centering
\end{figure}
The magnitude of the coupling terms are increasing with the rotation speed.
\begin{figure}[htbp]
\begin{subfigure}[c]{0.45\linewidth}
\includegraphics[width=\linewidth]{figs/plant_compare_rotating_speed_direct.pdf}
\caption{\label{fig:plant_compare_rotating_speed_direct} Direct Terms \(d_u/F_u\), \(d_v/F_v\)}
\end{subfigure}
\begin{subfigure}[c]{0.45\linewidth}
\includegraphics[width=\linewidth]{figs/plant_compare_rotating_speed_coupling.pdf}
\caption{\label{fig:plant_compare_rotating_speed_coupling} Coupling Terms \(d_v/F_u\), \(d_u/F_v\)}
\end{subfigure}
\caption{\label{fig:plant_compare_rotating_speed}Dynamics}
\centering
\includegraphics[scale=1]{figs/plant_compare_rotating_speed.pdf}
\caption{\label{fig:plant_compare_rotating_speed}Caption}
\end{figure}
\section{Integral Force Feedback}
\label{sec:orgaf500b0}
\label{sec:org2e8c85f}
\subsection{Control Schematic}
\label{sec:orgbd9f859}
\label{sec:org50b6359}
Force Sensors are added in series with the actuators as shown in Figure \ref{fig:system_iff}.
@@ -239,7 +247,7 @@ Force Sensors are added in series with the actuators as shown in Figure \ref{fig
\end{figure}
\subsection{Equations}
\label{sec:org48206d5}
\label{sec:org99c13a7}
The sensed forces are equal to:
\begin{equation}
\begin{bmatrix} f_{u} \\ f_{v} \end{bmatrix} =
@@ -270,9 +278,17 @@ Which then gives:
G_{fc} &= \left( 2 \xi \frac{s}{\omega_0} + 1 \right) \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)
\end{align}
\subsection{Plant Dynamics}
\label{sec:orgec8431d}
\label{sec:org1e476e3}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/plant_iff_compare_rotating_speed.pdf}
\caption{\label{fig:plant_iff_compare_rotating_speed}Figure caption}
\end{figure}
\subsection{Integral Force Feedback}
\label{sec:orga5d8887}
\begin{figure}[htbp]
\centering
@@ -280,19 +296,17 @@ Which then gives:
\caption{\label{fig:root_locus_pure_iff}Root Locus}
\end{figure}
\subsection{Physical Interpretation}
\label{sec:org159680e}
At low frequency, the gain is very large and thus no force is transmitted between the payload and the rotating stage.
This means that at low frequency, the system is decoupled (the force sensor removed) and thus the system is unstable.
\section{Integral Force Feedback with High Pass Filters}
\label{sec:org694707d}
\label{sec:org569b7db}
\subsection{Modification of the Control Low}
\label{sec:org931fb10}
\label{sec:org4d0c1ca}
\subsection{Close Loop Analysis}
\label{sec:org9de0aa7}
\subsection{Feedback Analysis}
\label{sec:org1f34d25}
\begin{figure}[htbp]
\centering
@@ -302,20 +316,19 @@ This means that at low frequency, the system is decoupled (the force sensor remo
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/root_locus_modified_iff_ter.pdf}
\includegraphics[scale=1]{figs/root_locus_modified_iff.pdf}
\caption{\label{fig:root_locus_modified_iff}Figure caption}
\end{figure}
\subsection{Optimal Cut-Off Frequency}
\label{sec:org9808de1}
\label{sec:org04a5aa4}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/root_locus_wi_modified_iff_bis.pdf}
\includegraphics[scale=1]{figs/root_locus_wi_modified_iff.pdf}
\caption{\label{fig:root_locus_wi_modified_iff}Figure caption}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/mod_iff_damping_wi.pdf}
@@ -323,31 +336,40 @@ This means that at low frequency, the system is decoupled (the force sensor remo
\end{figure}
\section{Integral Force Feedback with Parallel Springs}
\label{sec:orgd4915d5}
\label{sec:org03c54a4}
\subsection{Stiffness in Parallel with the Force Sensor}
\label{sec:orgc2d9221}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/rotating_xy_platform_springs.pdf}
\caption{\label{fig:rotating_xy_platform_springs}Figure caption}
\includegraphics[scale=1]{figs/system_parallel_springs.pdf}
\caption{\label{fig:system_parallel_springs}Figure caption}
\end{figure}
\subsection{Effect of the Parallel Stiffness on the Plant Dynamics}
\label{sec:org8097ba5}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/plant_iff_kp.pdf}
\caption{\label{fig:plant_iff_kp}Figure caption}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/root_locus_iff_kp.pdf}
\caption{\label{fig:root_locus_iff_kp}Figure caption}
\end{figure}
\subsection{Optimal Parallel Stiffness}
\label{sec:org1f2e167}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/root_locus_iff_kps.pdf}
\caption{\label{fig:root_locus_iff_kps}Figure caption}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/root_locus_iff_kp_ter.pdf}
\caption{\label{fig:root_locus_iff_kp_bis}Figure caption}
\end{figure}
\begin{figure}[htbp]
\centering
@@ -355,14 +377,23 @@ This means that at low frequency, the system is decoupled (the force sensor remo
\caption{\label{fig:root_locus_opt_gain_iff_kp}Figure caption}
\end{figure}
\section{Direct Velocity Feedback}
\label{sec:orgda2e325}
\subsection{Control Schematic}
\label{sec:org0e84009}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/plant_iff_compare_rotating_speed.pdf}
\caption{\label{fig:plant_iff_compare_rotating_speed}Figure caption}
\includegraphics[scale=1]{figs/system_dvf.pdf}
\caption{\label{fig:system_dvf}Figure caption}
\end{figure}
\section{Direct Velocity Feedback}
\label{sec:orgb0a5870}
\subsection{Equations}
\label{sec:org7cc244c}
\subsection{Relative Direct Velocity Feedback}
\label{sec:org668d842}
\begin{figure}[htbp]
\centering
@@ -370,8 +401,11 @@ This means that at low frequency, the system is decoupled (the force sensor remo
\caption{\label{fig:root_locus_dvf}Figure caption}
\end{figure}
\section{Comparison of the Proposed Active Damping Techniques}
\label{sec:org6097c1d}
\section{Comparison of the Proposed Active Damping Techniques for Rotating Positioning Stages}
\label{sec:orgade0126}
**
**
\begin{figure}[htbp]
\centering
@@ -379,25 +413,31 @@ This means that at low frequency, the system is decoupled (the force sensor remo
\caption{\label{fig:comp_root_locus}Figure caption}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/comp_compliance.pdf}
\caption{\label{fig:comp_compliance}Figure caption}
\end{figure}
\subsection{Transmissibility and Compliance}
\label{sec:org59b0db2}
\begin{figure}[htbp]
\begin{subfigure}[c]{0.45\linewidth}
\includegraphics[width=\linewidth]{figs/comp_compliance.pdf}
\caption{\label{fig:comp_compliance} Transmissibility}
\end{subfigure}
\begin{subfigure}[c]{0.45\linewidth}
\includegraphics[width=\linewidth]{figs/comp_transmissibility.pdf}
\caption{\label{fig:comp_transmissibility} Compliance}
\end{subfigure}
\caption{\label{fig:comp_active_damping}Comparison of the three proposed Active Damping Techniques}
\centering
\includegraphics[scale=1]{figs/comp_transmissibility.pdf}
\caption{\label{fig:comp_transmissibility}Figure caption}
\end{figure}
\section{Conclusion}
\label{sec:org1624a6b}
\label{sec:org4b70853}
\label{sec:conclusion}
\section*{Acknowledgment}
\label{sec:org1b29790}
\label{sec:org7708a14}
\bibliography{ref.bib}
\end{document}