% Created 2024-03-21 Thu 18:12
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\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}

\input{preamble.tex}
\bibliography{test-bench-apa.bib}
\author{Dehaeze Thomas}
\date{\today}
\title{Test Bench - Amplified Piezoelectric Actuator}
\hypersetup{
 pdfauthor={Dehaeze Thomas},
 pdftitle={Test Bench - Amplified Piezoelectric Actuator},
 pdfkeywords={},
 pdfsubject={},
 pdfcreator={Emacs 29.2 (Org mode 9.7)}, 
 pdflang={English}}
\usepackage{biblatex}

\begin{document}

\maketitle
\tableofcontents

\clearpage
\begin{figure}[htbp]
\centering
\includegraphics[scale=1,width=0.7\linewidth]{figs/test_apa_received.jpg}
\caption{\label{fig:test_apa_received}Picture of 5 out of the 7 received APA300ML}
\end{figure}

The first goal is to characterize the APA300ML in terms of:
\begin{itemize}
\item The, geometric features, electrical capacitance, stroke, hysteresis, spurious resonances.
This is performed in Section \ref{sec:test_apa_basic_meas}.
\item The dynamics from the generated DAC voltage (going to the voltage amplifiers and then applied on the actuator stacks) to the induced displacement, and to the measured voltage by the force sensor stack.
Also the ``actuator constant'' and ``sensor constant'' are identified.
This is done in Section \ref{sec:test_apa_dynamics}.
\item Compare the measurements with the Simscape models (2DoF, Super-Element) in order to tuned/validate the models.
This is explained in Section \ref{sec:test_apa_simscape}.
\end{itemize}

\begin{table}[htbp]
\caption{\label{tab:test_apa_section_matlab_code}Report sections and corresponding Matlab files}
\centering
\begin{tabularx}{0.6\linewidth}{lX}
\toprule
\textbf{Sections} & \textbf{Matlab File}\\
\midrule
Section \ref{sec:test_apa_basic_meas} & \texttt{test\_apa\_1\_basic\_meas.m}\\
Section \ref{sec:test_apa_dynamics} & \texttt{test\_apa\_2\_.m}\\
Section \ref{sec:test_apa_simscape} & \texttt{test\_apa\_3\_.m}\\
\bottomrule
\end{tabularx}
\end{table}
\chapter{First Basic Measurements}
\label{sec:test_apa_basic_meas}
Before using the measurement bench to characterize the APA300ML, first simple measurements are performed:
\begin{itemize}
\item Section \ref{sec:test_apa_geometrical_measurements}: the geometric tolerances of the interface planes are checked
\item Section \ref{sec:test_apa_electrical_measurements}: the capacitance of the piezoelectric stacks is measured
\item Section \ref{sec:test_apa_stroke_measurements}: the stroke of each APA is measured
\item Section \ref{sec:test_apa_spurious_resonances}: the ``spurious'' resonances of the APA are investigated
\end{itemize}
\section{Geometrical Measurements}
\label{sec:test_apa_geometrical_measurements}

To measure the flatness of the two mechanical interfaces of the APA300ML, a small measurement bench is installed on top of a metrology granite with very good flatness.

As shown in Figure \ref{fig:test_apa_flatness_setup}, the APA is fixed to a clamp while a measuring probe\footnote{Heidenhain MT25, specified accuracy of \(0.5\,\mu m\)} is used to measure the height of 4 points on each of the APA300ML interfaces.

From the X-Y-Z coordinates of the measured 8 points, the flatness is estimated by best fitting\footnote{The Matlab \texttt{fminsearch} command is used to fit the plane} a plane through all the points.

\begin{figure}[htbp]
\centering
\includegraphics[scale=1,width=0.4\linewidth]{figs/test_apa_flatness_setup.png}
\caption{\label{fig:test_apa_flatness_setup}Measurement setup for flatness estimation of the two mechanical interfaces}
\end{figure}

The measured flatness, summarized in Table \ref{tab:test_apa_flatness_meas}, are within the specifications.

\begin{table}[htbp]
\caption{\label{tab:test_apa_flatness_meas}Estimated flatness of the APA300ML interfaces}
\centering
\begin{tabularx}{0.3\linewidth}{Xc}
\toprule
 & \textbf{Flatness} \([\mu m]\)\\
\midrule
APA 1 & 8.9\\
APA 2 & 3.1\\
APA 3 & 9.1\\
APA 4 & 3.0\\
APA 5 & 1.9\\
APA 6 & 7.1\\
APA 7 & 18.7\\
\bottomrule
\end{tabularx}
\end{table}
\section{Electrical Measurements}
\label{sec:test_apa_electrical_measurements}

From the documentation of the APA300ML, the total capacitance of the three stacks should be between \(18\,\mu F\) and \(26\,\mu F\) with a nominal capacitance of \(20\,\mu F\).

The capacitance of the piezoelectric stacks found in the APA300ML have been measured with the LCR meter\footnote{LCR-819 from Gwinstek, specified accuracy of \(0.05\%\), measured frequency is set at \(1\,\text{kHz}\)} shown in Figure \ref{fig:test_apa_lcr_meter}.
The two stacks used as an actuator and the stack used as a force sensor are measured separately.

\begin{figure}[htbp]
\centering
\includegraphics[scale=1,width=0.6\linewidth]{figs/test_apa_lcr_meter.jpg}
\caption{\label{fig:test_apa_lcr_meter}LCR Meter used for the measurements}
\end{figure}

The measured capacitance are summarized in Table \ref{tab:test_apa_capacitance} and the average capacitance of one stack is \(\approx 5 \mu F\).
However, the measured capacitance of the stacks of ``APA 3'' is only half of the expected capacitance.
This may indicate a manufacturing defect.

The measured capacitance is found to be lower than the specified one.
This may be due to the fact that the manufacturer measures the capacitance with large signals (\(-20\,V\) to \(150\,V\)) while it was here measured with small signals.

\begin{table}[htbp]
\caption{\label{tab:test_apa_capacitance}Capacitance measured with the LCR meter. The excitation signal is a sinus at 1kHz}
\centering
\begin{tabularx}{0.5\linewidth}{lcc}
\toprule
 & \textbf{Sensor Stack} & \textbf{Actuator Stacks}\\
\midrule
APA 1 & 5.10 & 10.03\\
APA 2 & 4.99 & 9.85\\
APA 3 & 1.72 & 5.18\\
APA 4 & 4.94 & 9.82\\
APA 5 & 4.90 & 9.66\\
APA 6 & 4.99 & 9.91\\
APA 7 & 4.85 & 9.85\\
\bottomrule
\end{tabularx}
\end{table}
\section{Stroke Measurement}
\label{sec:test_apa_stroke_measurements}

The goal is here to verify that the stroke of the APA300ML is as specified in the datasheet.
To do so, one side of the APA is fixed to the granite, and a displacement probe\footnote{Millimar 1318 probe, specified linearity better than \(1\,\mu m\)} is located on the other side as shown in Figure \ref{fig:test_apa_stroke_bench}.

Then, the voltage across the two actuator stacks is varied from \(-20\,V\) to \(150\,V\) using a DAC and a voltage amplifier.
Note that the voltage is here slowly varied as the displacement probe has a very low measurement bandwidth (see Figure \ref{fig:test_apa_stroke_bench}, left).

\begin{figure}[htbp]
\centering
\includegraphics[scale=1,width=0.9\linewidth]{figs/test_apa_stroke_bench.jpg}
\caption{\label{fig:test_apa_stroke_bench}Bench to measured the APA stroke}
\end{figure}

The measured APA displacement is shown as a function of the applied voltage in Figure \ref{fig:test_apa_stroke_result}, right.

Typical hysteresis curves for piezoelectric stack actuators can be observed.
The measured stroke is approximately \(250\,\mu m\) when using only two of the three stacks, which is enough for the current application.
This is even above what is specified as the nominal stroke in the data-sheet (\(304\,\mu m\), therefore \(\approx 200\,\mu m\) if only two stacks are used).

It is clear from Figure \ref{fig:test_apa_stroke_result} that ``APA 3'' has an issue compared to the other units.
This confirms the abnormal electrical measurements made in Section \ref{sec:test_apa_electrical_measurements}.
This unit was send sent back to Cedrat and a new one was shipped back.
From now on, only the six APA that behave as expected will be used.

\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/test_apa_stroke_result.png}
\caption{\label{fig:test_apa_stroke_result}Generated voltage across the two piezoelectric stack actuators to estimate the stroke of the APA300ML (left). Measured displacement as a function of the applied voltage (right)}
\end{figure}
\section{Spurious resonances - APA\hfill{}\textsc{@philipp}}
\label{sec:test_apa_spurious_resonances}
\subsection{Introduction}

From a Finite Element Model of the struts, it have been found that three main resonances are foreseen to be problematic for the control of the APA300ML (Figure \ref{fig:apa_mode_shapes_ter}):
\begin{itemize}
\item Mode in X-bending at 189Hz
\item Mode in Y-bending at 285Hz
\item Mode in Z-torsion at 400Hz
\end{itemize}

\begin{figure}[htbp]
\centering
\includegraphics[scale=1,width=\linewidth]{figs/apa_mode_shapes.gif}
\caption{\label{fig:apa_mode_shapes_ter}Spurious resonances. a) X-bending mode at 189Hz. b) Y-bending mode at 285Hz. c) Z-torsion mode at 400Hz}
\end{figure}

These modes are present when flexible joints are fixed to the ends of the APA300ML.

In this section, we try to find the resonance frequency of these modes when one end of the APA is fixed and the other is free.

In the section \ref{sec:spurious_resonances_struts}, a similar measurement will be performed directly on the struts.
\subsection{Measurement Setup}

The measurement setup is shown in Figure \ref{fig:measurement_setup_torsion}.
A Laser vibrometer is measuring the difference of motion between two points.
The APA is excited with an instrumented hammer and the transfer function from the hammer to the measured rotation is computed.

\begin{note}
The instrumentation used are:
\begin{itemize}
\item Laser Doppler Vibrometer Polytec OFV512
\item Instrumented hammer
\end{itemize}
\end{note}

\begin{figure}[htbp]
\centering
\includegraphics[scale=1,width=0.7\linewidth]{figs/measurement_setup_torsion.jpg}
\caption{\label{fig:measurement_setup_torsion}Measurement setup with a Laser Doppler Vibrometer and one instrumental hammer}
\end{figure}
\subsection{X-Bending Mode}

The vibrometer is setup to measure the X-bending motion is shown in Figure \ref{fig:measurement_setup_X_bending}.
The APA is excited with an instrumented hammer having a solid metallic tip.
The impact point is on the back-side of the APA aligned with the top measurement point.

\begin{figure}[htbp]
\centering
\includegraphics[scale=1,width=0.7\linewidth]{figs/measurement_setup_X_bending.jpg}
\caption{\label{fig:measurement_setup_X_bending}X-Bending measurement setup}
\end{figure}

The data is loaded.
The configuration (Sampling time and windows) for \texttt{tfestimate} is done:
The transfer function from the input force to the output ``rotation'' (difference between the two measured distances).
The result is shown in Figure \ref{fig:apa300ml_meas_freq_bending_x}.

The can clearly observe a nice peak at 280Hz, and then peaks at the odd ``harmonics'' (third ``harmonic'' at 840Hz, and fifth ``harmonic'' at 1400Hz).
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/apa300ml_meas_freq_bending_x.png}
\caption{\label{fig:apa300ml_meas_freq_bending_x}Obtained FRF for the X-bending}
\end{figure}

Then the APA is in the ``free-free'' condition, this bending mode is foreseen to be at 200Hz (Figure \ref{fig:apa_mode_shapes_ter}).
We are here in the ``fixed-free'' condition.
If we consider that we therefore double the stiffness associated with this mode, we should obtain a resonance a factor \(\sqrt{2}\) higher than 200Hz which is indeed 280Hz.
Not sure this reasoning is correct though.
\subsection{Y-Bending Mode}

The setup to measure the Y-bending is shown in Figure \ref{fig:measurement_setup_Y_bending}.

The impact point of the instrumented hammer is located on the back surface of the top interface (on the back of the 2 measurements points).

\begin{figure}[htbp]
\centering
\includegraphics[scale=1,width=0.7\linewidth]{figs/measurement_setup_Y_bending.jpg}
\caption{\label{fig:measurement_setup_Y_bending}Y-Bending measurement setup}
\end{figure}

The data is loaded, and the transfer function from the force to the measured rotation is computed.
The results are shown in Figure \ref{fig:apa300ml_meas_freq_bending_y}.
The main resonance is at 412Hz, and we also see the third ``harmonic'' at 1220Hz.

\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/apa300ml_meas_freq_bending_y.png}
\caption{\label{fig:apa300ml_meas_freq_bending_y}Obtained FRF for the Y-bending}
\end{figure}

We can apply the same reasoning as in the previous section and estimate the mode to be a factor \(\sqrt{2}\) higher than the mode estimated in the ``free-free'' condition.
We would obtain a mode at 403Hz which is very close to the one estimated here.
\subsection{Z-Torsion Mode}

Finally, we measure the Z-torsion resonance as shown in Figure \ref{fig:measurement_setup_torsion_bis}.

The excitation is shown on the other side of the APA, on the side to excite the torsion motion.

\begin{figure}[htbp]
\centering
\includegraphics[scale=1,width=0.7\linewidth]{figs/measurement_setup_torsion_bis.jpg}
\caption{\label{fig:measurement_setup_torsion_bis}Z-Torsion measurement setup}
\end{figure}

The data is loaded, and the transfer function computed.
The results are shown in Figure \ref{fig:apa300ml_meas_freq_torsion_z}.
We observe a first peak at 267Hz, which corresponds to the X-bending mode that was measured at 280Hz.
And then a second peak at 415Hz, which corresponds to the X-bending mode that was measured at 412Hz.
A third mode at 800Hz could correspond to this torsion mode.

\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/apa300ml_meas_freq_torsion_z.png}
\caption{\label{fig:apa300ml_meas_freq_torsion_z}Obtained FRF for the Z-torsion}
\end{figure}

In order to verify that, the APA is excited on the top part such that the torsion mode should not be excited.
The two FRF are compared in Figure \ref{fig:apa300ml_meas_freq_torsion_z_comp}.
It is clear that the first two modes does not correspond to the torsional mode.
Maybe the resonance at 800Hz, or even higher resonances. It is difficult to conclude here.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/apa300ml_meas_freq_torsion_z_comp.png}
\caption{\label{fig:apa300ml_meas_freq_torsion_z_comp}Obtained FRF for the Z-torsion}
\end{figure}
\subsection{Compare}
The three measurements are shown in Figure \ref{fig:apa300ml_meas_freq_compare}.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/apa300ml_meas_freq_compare.png}
\caption{\label{fig:apa300ml_meas_freq_compare}Obtained FRF - Comparison}
\end{figure}
\subsection{Conclusion}

When two flexible joints are fixed at each ends of the APA, the APA is mostly in a free/free condition in terms of bending/torsion (the bending/torsional stiffness of the joints being very small).

In the current tests, the APA are in a fixed/free condition.
Therefore, it is quite obvious that we measured higher resonance frequencies than what is foreseen for the struts.
It is however quite interesting that there is a factor \(\approx \sqrt{2}\) between the two (increased of the stiffness by a factor 2?).

\begin{table}[htbp]
\caption{\label{tab:apa300ml_measured_modes_freq}Measured frequency of the modes}
\centering
\begin{tabularx}{0.7\linewidth}{Xcc}
\toprule
\textbf{Mode} & \textbf{FEM - Strut mode} & \textbf{Measured Frequency}\\
\midrule
X-Bending & 189Hz & 280Hz\\
Y-Bending & 285Hz & 410Hz\\
Z-Torsion & 400Hz & 800Hz?\\
\bottomrule
\end{tabularx}
\end{table}
\chapter{Dynamical measurements - APA}
\label{sec:test_apa_dynamics}

After the basic measurements on the APA were performed in Section \ref{sec:test_apa_basic_meas}, a new test bench is used to better characterize the APA.

This test bench is shown in Figure \ref{fig:test_bench_apa} and consists of the APA300ML fixed on one end to the fixed granite, and on the other end to the 5kg granite vertically guided with an air bearing.
An encoder is used to measure the relative motion between the two granites (i.e. the displacement of the APA).

\begin{figure}
\begin{subfigure}{0.3\textwidth}
\begin{center}
\includegraphics[scale=1,height=8cm]{figs/test_apa_bench_picture.jpg}
\end{center}
\subcaption{\label{fig:test_apa_bench_picture}Picture of the test bench}
\end{subfigure}
\begin{subfigure}{0.69\textwidth}
\begin{center}
\includegraphics[scale=1,height=8cm]{figs/test_apa_bench_picture_encoder.jpg}
\end{center}
\subcaption{\label{fig:test_apa_bench_picture_encoder}Zoom on the APA with the encoder}
\end{subfigure}
\caption{\label{fig:test_bench_apa}Test bench used to characterize the APA300ML}
\end{figure}

The bench is schematically shown in Figure \ref{fig:test_apa_schematic} and the signal used are summarized in Table \ref{tab:test_apa_variables}.

\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=1]{figs/test_apa_schematic.png}
\caption{\label{fig:test_apa_schematic}Schematic of the Test Bench}
\end{figure}

\begin{table}[htbp]
\caption{\label{tab:test_apa_variables}Variables used during the measurements}
\centering
\begin{tabularx}{0.6\linewidth}{cXc}
\toprule
\textbf{Variable} & \textbf{Description} & \textbf{Unit}\\
\midrule
\(u\) & Output DAC Voltage & \(V\)\\
\(V_a\) & Output Amplifier Voltage & \(V\)\\
\(V_s\) & Measured Stack Voltage (ADC) & \(V\)\\
\(d_e\) & Encoder Measurement & \(m\)\\
\bottomrule
\end{tabularx}
\end{table}

This bench will be used to:
\begin{itemize}
\item measure the dynamics of the APA (from \(V_a\) to \(d_e\) and \(d_a\) in Section \ref{ssec:test_apa_meas_frf_disp}, and from \(V_a\) to \(V_s\) in section \ref{ssec:test_apa_meas_frf_force})
\item estimate the added damping using Integral Force Feedback (Section \ref{ssec:test_apa_iff_locus})
\end{itemize}

These measurements will also be used to tune the model of the APA in Section \ref{sec:test_apa_simscape}.
\section{Hysteresis}
\label{ssec:test_apa_hysteresis}

As the payload is vertically guided without friction, the hysteresis of the APA can be estimated from the motion of the payload.

A quasi static sinusoidal excitation \(V_a\) with an offset of \(65\,V\) (halfway between \(-20\,V\) and \(150\,V\)), and an amplitude varying from \(4\,V\) up to \(80\,V\).

For each excitation amplitude, the vertical displacement \(d_e\) of the mass is measured and displayed as a function of the applied voltage..

The measured displacements as a function of the output voltages are shown in Figure \ref{fig:test_apa_meas_hysteresis}.
It is interesting to see that the hysteresis is increasing with the excitation amplitude.

\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/test_apa_meas_hysteresis.png}
\caption{\label{fig:test_apa_meas_hysteresis}Obtained hysteresis curves (displacement as a function of applied voltage) for multiple excitation amplitudes}
\end{figure}
\section{Axial stiffness}
\label{ssec:test_apa_stiffness}

In order to estimate the stiffness of the APA, a weight with known mass \(m_a = 6.4\,\text{kg}\) is added on top of the suspended granite and the deflection \(d_e\) is measured using the encoder.

The APA stiffness can then be estimated from equation \eqref{eq:test_apa_stiffness}.

\begin{equation} \label{eq:test_apa_stiffness}
  k_{\text{apa}} = \frac{m_a g}{\Delta d_e}
\end{equation}

The measured displacement \(d_e\) as a function of time is shown in Figure \ref{fig:test_apa_meas_stiffness_time}.
It can be seen that there are some drifts in the measured displacement (probably due to piezoelectric creep) and the that displacement does not come back to the initial position after the mass is removed (probably due to piezoelectric hysteresis).
These two effects induce some uncertainties in the measured stiffness.

\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/test_apa_meas_stiffness_time.png}
\caption{\label{fig:test_apa_meas_stiffness_time}Measured displacement when adding the mass (at \(t \approx 3\,s\)) and removing the mass(at \(t \approx 13\,s\))}
\end{figure}

The stiffnesses are computed for all the APA from the two displacements \(d_1\) and \(d_2\) (see Figure \ref{fig:test_apa_meas_stiffness_time}) leading to two stiffness estimations \(k_1\) and \(k_2\).
These estimated stiffnesses are summarized in Table \ref{tab:test_apa_measured_stiffnesses} and are found to be close to the nominal stiffness \(k = 1.8\,N/\mu m\) found in the APA300ML manual.

\begin{table}[htbp]
\caption{\label{tab:test_apa_measured_stiffnesses}Measured stiffnesses (in \(N/\mu m\))}
\centering
\begin{tabularx}{0.2\linewidth}{ccc}
\toprule
APA & \(k_1\) & \(k_2\)\\
\midrule
1 & 1.68 & 1.9\\
2 & 1.69 & 1.9\\
4 & 1.7 & 1.91\\
5 & 1.7 & 1.93\\
6 & 1.7 & 1.92\\
8 & 1.73 & 1.98\\
\bottomrule
\end{tabularx}
\end{table}

The stiffness can also be computed using equation \eqref{eq:test_apa_res_freq} by knowing the main vertical resonance frequency \(\omega_z \approx 94\,\text{Hz}\) (estimated by the dynamical measurements shown in section \ref{ssec:test_apa_meas_frf_disp}) and the suspended mass \(m_{\text{sus}} = 5.7\,\text{kg}\).

\begin{equation} \label{eq:test_apa_res_freq}
\omega_z = \sqrt{\frac{k}{m_{\text{sus}}}}
\end{equation}

The obtain stiffness is \(k \approx 2\,N/\mu m\) which is close to the values found in the documentation and by the ``static deflection'' method.


However, changes in the electrical impedance connected to the piezoelectric stacks impacts the mechanical compliance (or stiffness) of the piezoelectric stack \cite[chap. 2]{reza06_piezoel_trans_vibrat_contr_dampin}.

To estimate this effect, the stiffness of the APA if measured using the ``static deflection'' method in two cases:
\begin{itemize}
\item \(k_{\text{os}}\): piezoelectric stacks left unconnected (or connect to the high impedance ADC)
\item \(k_{\text{sc}}\): piezoelectric stacks short circuited (or connected to the voltage amplifier with small output impedance)
\end{itemize}

The open-circuit stiffness is estimated at \(k_{\text{oc}} \approx 2.3\,N/\mu m\) and the closed-circuit stiffness \(k_{\text{sc}} \approx 1.7\,N/\mu m\).
\section{Dynamics}
\label{ssec:test_apa_meas_dynamics}

In this section, the dynamics of the system from the excitation voltage \(u\) to encoder measured displacement \(d_e\)  and to the force sensor voltage \(V_s\) is identified.

The obtained transfer functions for the 6 APA between the excitation voltage \(u\) and the encoder displacement \(d_e\) are shown in Figure \ref{fig:test_apa_frf_encoder}.
The obtained transfer functions are close to a mass-spring-damper system.
The following can be observed:
\begin{itemize}
\item A ``stiffness line'' indicating a static gain equal to \(\approx -17\,\mu m/V\).
The minus sign comes from the fact that an increase in voltage stretches the piezoelectric stack that then reduces the height of the APA
\item A lightly damped resonance at \(95\,\text{Hz}\)
\item A ``mass line'' up to \(\approx 800\,\text{Hz}\), above which some resonances appear
\end{itemize}

\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/test_apa_frf_encoder.png}
\caption{\label{fig:test_apa_frf_encoder}Estimated Frequency Response Function from generated voltage \(u\) to the encoder displacement \(d_e\) for the 6 APA300ML}
\end{figure}

The dynamics from \(u\) to the measured voltage across the sensor stack \(V_s\) is also identified and shown in Figure \ref{fig:test_apa_frf_force}.

A lightly damped resonance is observed at \(95\,\text{Hz}\) and a lightly damped anti-resonance at \(41\,\text{Hz}\).
No additional resonances is present up to at least \(2\,\text{kHz}\) indicating at Integral Force Feedback can be applied without stability issues from high frequency flexible modes.

As illustrated by the Root Locus, the poles of the closed-loop system converges to the zeros of the open-loop plant.
Suppose that a controller with a very high gain is implemented such that the voltage \(V_s\) across the sensor stack is zero.
In that case, because of the very high controller gain, no stress and strain is present on the sensor stack (and on the actuator stacks are well, as they are both in series).
Such closed-loop system would therefore virtually corresponds to a system for which the piezoelectric stacks have been removed and just the mechanical shell is kept.
From this analysis, the axial stiffness of the shell can be estimated to be \(k_{\text{shell}} = 5.7 \cdot (2\pi \cdot 41)^2 = 0.38\,N/\mu m\).
Such reasoning can lead to very interesting insight into the system just from an open-loop identification.

\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/test_apa_frf_force.png}
\caption{\label{fig:test_apa_frf_force}Estimated Frequency Response Function from generated voltage \(u\) to the sensor stack voltage \(V_s\) for the 6 APA300ML}
\end{figure}

All the identified dynamics of the six APA300ML (both when looking at the encoder in Figure \ref{fig:test_apa_frf_encoder} and at the force sensor in Figure \ref{fig:test_apa_frf_force}) are almost identical, indicating good manufacturing repeatability for the piezoelectric stacks and the mechanical lever.
\section{Effect of the resistor on the IFF Plant}
\label{ssec:test_apa_resistance_sensor_stack}

A resistor \(R \approx 80.6\,k\Omega\) is added in parallel with the sensor stack which has the effect to form a high pass filter with the capacitance of the stack.

As explain before, this is done for two reasons:
\begin{enumerate}
\item Limit the voltage offset due to the input bias current of the ADC
\item Limit the low frequency gain
\end{enumerate}

The (low frequency) transfer function from \(u\) to \(V_s\) with and without this resistor have been measured and are compared in Figure \ref{fig:test_apa_effect_resistance}.
It is confirmed that the added resistor as the effect of adding an high pass filter with a cut-off frequency of \(\approx 0.35\,\text{Hz}\).

\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/test_apa_effect_resistance.png}
\caption{\label{fig:test_apa_effect_resistance}Transfer function from u to \(V_s\) with and without the resistor \(R\) in parallel with the piezoelectric stack used as the force sensor}
\end{figure}
\section{Integral Force Feedback}
\label{ssec:test_apa_iff_locus}

This test bench can also be used to estimate the damping added by the implementation of an Integral Force Feedback strategy.

First, the transfer function \eqref{eq:test_apa_iff_manual_fit} is manually tuned to match the identified dynamics from generated voltage \(u\) to the measured sensor stack voltage \(V_s\) in Section \ref{ssec:test_apa_meas_dynamics}.

The obtained parameter values are \(\omega_{\textsc{hpf}} = 0.4\, \text{Hz}\), \(\omega_{z} = 42.7\, \text{Hz}\), \(\xi_{z} = 0.4\,\%\), \(\omega_{p} = 95.2\, \text{Hz}\), \(\xi_{p} = 2\,\%\) and \(g_0 = 0.64\).

\begin{equation} \label{eq:test_apa_iff_manual_fit}
G_{\textsc{iff},m}(s) = g_0 \cdot \frac{1 + 2 \xi_z \frac{s}{\omega_z} + \frac{s^2}{\omega_z^2}}{1 + 2 \xi_p \frac{s}{\omega_p} + \frac{s^2}{\omega_p^2}} \cdot \frac{s}{\omega_{\textsc{hpf}} + s}
\end{equation}

The comparison between the identified plant and the manually tuned transfer function is done in Figure \ref{fig:test_apa_iff_plant_comp_manual_fit}.

\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/test_apa_iff_plant_comp_manual_fit.png}
\caption{\label{fig:test_apa_iff_plant_comp_manual_fit}Identified IFF plant and manually tuned model of the plant (a time delay of \(200\,\mu s\) is added to the model of the plant to better match the identified phase)}
\end{figure}

The implemented Integral Force Feedback Controller transfer function is shown in equation \eqref{eq:test_apa_Kiff_formula}.
It contains an high pass filter (cut-off frequency of \(2\,\text{Hz}\)) to limit the low frequency gain, a low pass filter to add integral action above \(20\,\text{Hz}\), a second low pass filter to add robustness to high frequency resonances and a tunable gain \(g\).

\begin{equation} \label{eq:test_apa_Kiff_formula}
K_{\textsc{iff}}(s) = -10 \cdot g \cdot \frac{s}{s + 2\pi \cdot 2} \cdot \frac{1}{1 + 2\pi \cdot 20} \cdot \frac{1}{s + 2\pi\cdot 2000}
\end{equation}

To estimate how the dynamics of the APA changes when the Integral Force Feedback controller is implemented, the test bench shown in Figure \ref{fig:test_apa_iff_schematic} is used.
The transfer function from the ``damped'' plant input \(u\prime\) to the encoder displacement \(d_e\) is identified for several IFF controller gains \(g\).

\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/test_apa_iff_schematic.png}
\caption{\label{fig:test_apa_iff_schematic}Figure caption}
\end{figure}

The identified dynamics are then fitted by second order transfer functions using the ``Vector Fitting'' toolbox \cite{gustavsen99_ration_approx_frequen_domain_respon}.
The comparison between the identified damped dynamics and the fitted second order transfer functions is done in Figure \ref{fig:test_apa_identified_damped_plants} for different gains \(g\).
It is clear that large amount of damping is added when the gain is increased and that the frequency of the pole is shifted to lower frequencies.

\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/test_apa_identified_damped_plants.png}
\caption{\label{fig:test_apa_identified_damped_plants}Identified dynamics (solid lines) and fitted transfer functions (dashed lines) from \(u\prime\) to \(d_e\) for different IFF gains}
\end{figure}

The evolution of the pole in the complex plane as a function of the controller gain \(g\) (i.e. the ``root locus'') is computed:
\begin{itemize}
\item using the IFF plant model \eqref{eq:test_apa_iff_manual_fit} and the implemented controller \eqref{eq:test_apa_Kiff_formula}
\item from the fitted transfer functions of the damped plants experimentally identified for several controller gains
\end{itemize}

The two obtained root loci are compared in Figure \ref{fig:test_apa_iff_root_locus} and are in good agreement considering that the damped plants were only fitted using a second order transfer function.

\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/test_apa_iff_root_locus.png}
\caption{\label{fig:test_apa_iff_root_locus}Root Locus of the APA300ML with Integral Force Feedback - Comparison between the computed root locus from the plant model (black line) and the root locus estimated from the damped plant pole identification (colorful crosses)}
\end{figure}
\section{Conclusion}

\begin{important}
So far, all the measured FRF are showing the dynamical behavior that was expected.
\end{important}
\chapter{Test Bench APA300ML - Simscape Model}
\label{sec:test_apa_simscape}
In this section, a simscape model (Figure \ref{fig:model_bench_apa}) of the measurement bench is used to compare the model of the APA with the measured FRF.

After the transfer functions are extracted from the model (Section \ref{sec:simscape_bench_apa_first_id}), the comparison of the obtained dynamics with the measured FRF will permit to:
\begin{enumerate}
\item Estimate the ``actuator constant'' and ``sensor constant'' (Section \ref{sec:simscape_bench_apa_id_constants})
\item ``Actuator constant'': Gain from the applied voltage \(V_a\) to the generated Force \(F_a\)
\item ``Sensor constant'': Gain from the sensor stack strain \(\delta L\) to the generated voltage \(V_s\)
\item Tune the model of the APA to match the measured dynamics (Section \ref{sec:simscape_bench_apa_tune_2dof_model})
\end{enumerate}

\begin{figure}[htbp]
\centering
\includegraphics[scale=1,width=0.5\linewidth]{figs/model_bench_apa.png}
\caption{\label{fig:model_bench_apa}Screenshot of the Simscape model}
\end{figure}
\section{First Identification}
\label{sec:simscape_bench_apa_first_id}

The APA is first initialized with default parameters:
The transfer function from excitation voltage \(V_a\) (before the amplification of \(20\) due to the PD200 amplifier) to:
\begin{enumerate}
\item the sensor stack voltage \(V_s\)
\item the measured displacement by the encoder \(d_e\)
\end{enumerate}

The obtain dynamics are shown in Figure \ref{fig:apa_model_bench_bode_vs} and \ref{fig:apa_model_bench_bode_dl_z}.
It can be seen that:
\begin{itemize}
\item the shape of these bode plots are very similar to the one measured in Section \ref{sec:dynamical_meas_apa} expect from a change in gain and exact location of poles and zeros
\item there is a sign error for the transfer function from \(V_a\) to \(V_s\).
This will be corrected by taking a negative ``sensor gain''.
\item the low frequency zero of the transfer function from \(V_a\) to \(V_s\) is minimum phase as expected.
The measured FRF are showing non-minimum phase zero, but it is most likely due to measurements artifacts.
\end{itemize}

\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/apa_model_bench_bode_vs.png}
\caption{\label{fig:apa_model_bench_bode_vs}Bode plot of the transfer function from \(V_a\) to \(V_s\)}
\end{figure}

\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/apa_model_bench_bode_dl_z.png}
\caption{\label{fig:apa_model_bench_bode_dl_z}Bode plot of the transfer function from \(V_a\) to \(d_L\) and to \(z\)}
\end{figure}
\section{Identify Sensor/Actuator constants and compare with measured FRF}
\label{sec:simscape_bench_apa_id_constants}
\subsection{How to identify these constants?}
\paragraph{Piezoelectric Actuator Constant}

Using the measurement test-bench, it is rather easy the determine the static gain between the applied voltage \(V_a\) to the induced displacement \(d\).
\begin{equation}
  d = g_{d/V_a} \cdot V_a
\end{equation}

Using the Simscape model of the APA, it is possible to determine the static gain between the actuator force \(F_a\) to the induced displacement \(d\):
\begin{equation}
  d = g_{d/F_a} \cdot F_a
\end{equation}

From the two gains, it is then easy to determine \(g_a\):
\begin{equation} \label{eq:actuator_constant_formula}
  \boxed{g_a = \frac{F_a}{V_a} = \frac{F_a}{d} \cdot \frac{d}{V_a} = \frac{g_{d/V_a}}{g_{d/F_a}}}
\end{equation}
\paragraph{Piezoelectric Sensor Constant}

Similarly, it is easy to determine the gain from the excitation voltage \(V_a\) to the voltage generated by the sensor stack \(V_s\):
\begin{equation}
  V_s = g_{V_s/V_a} V_a
\end{equation}

Note here that there is an high pass filter formed by the piezoelectric capacitor and parallel resistor.

The gain can be computed from the dynamical identification and taking the gain at the wanted frequency (above the first resonance).

Using the simscape model, compute the gain at the same frequency from the actuator force \(F_a\) to the strain of the sensor stack \(dl\):
\begin{equation}
  dl = g_{dl/F_a} F_a
\end{equation}

Then, the ``sensor'' constant is:
\begin{equation} \label{eq:sensor_constant_formula}
  \boxed{g_s = \frac{V_s}{dl} = \frac{V_s}{V_a} \cdot \frac{V_a}{F_a} \cdot \frac{F_a}{dl} = \frac{g_{V_s/V_a}}{g_a \cdot g_{dl/F_a}}}
\end{equation}
\subsection{Identification Data}
Let's load the measured FRF from the DAC voltage to the measured encoder and to the sensor stack voltage.
\subsection{2DoF APA}
\paragraph{2DoF APA}
Let's initialize the APA as a 2DoF model with unity sensor and actuator gains.
\paragraph{Identification without actuator or sensor constants}
The transfer function from \(V_a\) to \(V_s\), \(d_e\) and \(d_a\) is identified.
\paragraph{Actuator Constant}
Then, the actuator constant can be computed as shown in Eq. \eqref{eq:actuator_constant_formula} by dividing the measured DC gain of the transfer function from \(V_a\) to \(d_e\) by the estimated DC gain of the transfer function from \(V_a\) (in truth the actuator force called \(F_a\)) to \(d_e\) using the Simscape model.
\begin{verbatim}
ga = -32.2 [N/V]
\end{verbatim}
\paragraph{Sensor Constant}
Similarly, the sensor constant can be estimated using Eq. \eqref{eq:sensor_constant_formula}.

\begin{verbatim}
gs = 0.088 [V/m]
\end{verbatim}
\paragraph{Comparison}
Let's now initialize the APA with identified sensor and actuator constant:
And identify the dynamics with included constants.
The transfer functions from \(V_a\) to \(d_e\) are compared in Figure \ref{fig:apa_act_constant_comp} and the one from \(V_a\) to \(V_s\) are compared in Figure \ref{fig:apa_sens_constant_comp}.

\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/apa_act_constant_comp.png}
\caption{\label{fig:apa_act_constant_comp}Comparison of the experimental data and Simscape model (\(V_a\) to \(d_e\))}
\end{figure}

\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/apa_sens_constant_comp.png}
\caption{\label{fig:apa_sens_constant_comp}Comparison of the experimental data and Simscape model (\(V_a\) to \(V_s\))}
\end{figure}

\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/apa_comp_model_frf.png}
\label{fig:apa_comp_model_frf}
\end{figure}


\begin{important}
The ``actuator constant'' and ``sensor constant'' can indeed be identified using this test bench.
After identifying these constants, the 2DoF model shows good agreement with the measured dynamics.
\end{important}
\subsection{Flexible APA}
In this section, the sensor and actuator ``constants'' are also estimated for the flexible model of the APA.
\paragraph{Flexible APA}
The Simscape APA model is initialized as a flexible one with unity ``constants''.
\paragraph{Identification without actuator or sensor constants}
The dynamics from \(V_a\) to \(V_s\), \(d_e\) and \(d_a\) is identified.
\paragraph{Actuator Constant}
Then, the actuator constant can be computed as shown in Eq. \eqref{eq:actuator_constant_formula}:
\begin{verbatim}
ga = 23.5 [N/V]
\end{verbatim}
\paragraph{Sensor Constant}
\begin{verbatim}
gs = -4839841.756 [V/m]
\end{verbatim}
\paragraph{Comparison}
Let's now initialize the flexible APA with identified sensor and actuator constant:
And identify the dynamics with included constants.
The obtained dynamics is compared with the measured one in Figures \ref{fig:apa_act_constant_comp_flex} and \ref{fig:apa_sens_constant_comp_flex}.

\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/apa_act_constant_comp_flex.png}
\caption{\label{fig:apa_act_constant_comp_flex}Comparison of the experimental data and Simscape model (\(u\) to \(d\mathcal{L}_m\))}
\end{figure}

\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/apa_sens_constant_comp_flex.png}
\caption{\label{fig:apa_sens_constant_comp_flex}Comparison of the experimental data and Simscape model (\(u\) to \(\tau_m\))}
\end{figure}

\begin{important}
The flexible model is a bit ``soft'' as compared with the experimental results.
\end{important}
\section{Optimize 2-DoF model to fit the experimental Data}
\label{sec:simscape_bench_apa_tune_2dof_model}
The parameters of the 2DoF model presented in Section \ref{sec:apa_2dof_model} are now optimize such that the model best matches the measured FRF.

After optimization, the following parameters are used:
The dynamics is identified using the Simscape model and compared with the measured FRF in Figure \ref{fig:comp_apa_plant_after_opt}.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/comp_apa_plant_after_opt.png}
\caption{\label{fig:comp_apa_plant_after_opt}Comparison of the measured FRF and the optimized model}
\end{figure}

\begin{important}
The tuned 2DoF is very well representing the (axial) dynamics of the APA.
\end{important}
\chapter{Conclusion}
\label{sec:test_apa_conclusion}
\printbibliography[heading=bibintoc,title={Bibliography}]
\end{document}
