Christophe's reviews

test-bench-apa.org

@@ -118,16 +118,41 @@ Verify that everything interesting to say about that is either done before in th
 CLOSED: [2024-04-04 Thu 10:42]
 
 - State "CANC"       from "TODO"       [2024-04-04 Thu 10:42]
+* Glossary and Acronyms - Tables                                      :ignore:
+
+#+name: glossary
+| label | name                    | description                                 |
+|-------+-------------------------+---------------------------------------------|
+| psdx  | \ensuremath{\Phi_{x}}   | Power spectral density of signal $x$        |
+| asdx  | \ensuremath{\Gamma_{x}} | Amplitude spectral density of signal $x$    |
+| cpsx  | \ensuremath{\Phi_{x}}   | Cumulative Power Spectrum of signal $x$     |
+| casx  | \ensuremath{\Gamma_{x}} | Cumulative Amplitude Spectrum of signal $x$ |
+
+#+name: acronyms
+| key    | abbreviation | full form                                      |
+|--------+--------------+------------------------------------------------|
+| haclac | HAC-LAC      | High Authority Control - Low Authority Control |
+| hac    | HAC          | High Authority Control                         |
+| lac    | LAC          | Low Authority Control                          |
+| nass   | NASS         | Nano Active Stabilization System               |
+| asd    | ASD          | Amplitude Spectral Density                     |
+| psd    | PSD          | Power Spectral Density                         |
+| cps    | CPS          | Cumulative Power Spectrum                      |
+| cas    | CAS          | Cumulative Amplitude Spectrum                  |
+| frf    | FRF          | Frequency Response Function                    |
+| iff    | IFF          | Integral Force Feedback                        |
+| rdc    | RDC          | Relative Damping Control                       |
+| drga   | DRGA         | Dynamical Relative Gain Array                  |
+| hpf    | HPF          | high-pass filter                               |
+| lpf    | LPF          | low-pass filter                                |
+| dof    | DoF          | Degrees of Freedom                             |
+
 * Introduction                                                        :ignore:
 
 In this chapter, the goal is to make sure that the received APA300ML (shown in Figure ref:fig:test_apa_received) are complying with the requirements and that dynamical models of the actuator are well representing its dynamics.
 
-#+name: fig:test_apa_received
+In section ref:sec:test_apa_basic_meas, the mechanical tolerances of the APA300ML interfaces are checked together with the electrical properties of the piezoelectric stacks, the achievable stroke.
-#+attr_latex: :width 0.7\linewidth
+Flexible modes of the APA300ML which were estimated using a finite element model are compared with measurements.
-#+caption: Picture of 5 out of the 7 received APA300ML
-[[file:figs/test_apa_received.jpg]]
-
-In section ref:sec:test_apa_basic_meas, the mechanical tolerances of the APA300ML interfaces are checked together with the electrical properties of the piezoelectric stacks, the achievable stroke. Flexible modes of the APA300ML are computed with a finite element model and compared with measurements.
 
 Using a dedicated test bench, dynamical measurements are performed (Section ref:sec:test_apa_dynamics).
 The dynamics from the generated DAC voltage (going through the voltage amplifier and then to two actuator stacks) to the induced axial displacement and to the measured voltage across the force sensor stack are estimated.
@@ -135,21 +160,26 @@ Integral Force Feedback is experimentally applied and the damped plants are esti
 
 Two different models of the APA300ML are then presented.
 First, in Section ref:sec:test_apa_model_2dof, a two degrees of freedom model is presented, tuned and compared with the measured dynamics.
-This model is proven to accurately simulate the APA300ML's axial dynamics.
+This model is proven to accurately represents the APA300ML's axial dynamics while having low complexity.
 
 Then, in Section ref:sec:test_apa_model_flexible, a /super element/ of the APA300ML is extracted using a finite element model and imported in Simscape.
 This more complex model is also shown to well capture the axial dynamics of the APA300ML.
 
-#+name: tab:test_apa_section_matlab_code
+#+name: fig:test_apa_received
-#+caption: Report sections and corresponding Matlab files
+#+attr_latex: :width 0.7\linewidth
-#+attr_latex: :environment tabularx :width 0.6\linewidth :align lX
+#+caption: Picture of 5 out of the 7 received APA300ML
-#+attr_latex: :center t :booktabs t
+[[file:figs/test_apa_received.jpg]]
-| *Sections*                              | *Matlab File*                 |
+
-|-----------------------------------------+-------------------------------|
+# #+name: tab:test_apa_section_matlab_code
-| Section ref:sec:test_apa_basic_meas     | =test_apa_1_basic_meas.m=     |
+# #+caption: Report sections and corresponding Matlab files
-| Section ref:sec:test_apa_dynamics       | =test_apa_2_dynamics.m=       |
+# #+attr_latex: :environment tabularx :width 0.6\linewidth :align lX
-| Section ref:sec:test_apa_model_2dof     | =test_apa_3_model_2dof.m=     |
+# #+attr_latex: :center t :booktabs t
-| Section ref:sec:test_apa_model_flexible | =test_apa_4_model_flexible.m= |
+# | *Sections*                              | *Matlab File*                 |
+# |-----------------------------------------+-------------------------------|
+# | Section ref:sec:test_apa_basic_meas     | =test_apa_1_basic_meas.m=     |
+# | Section ref:sec:test_apa_dynamics       | =test_apa_2_dynamics.m=       |
+# | Section ref:sec:test_apa_model_2dof     | =test_apa_3_model_2dof.m=     |
+# | Section ref:sec:test_apa_model_flexible | =test_apa_4_model_flexible.m= |
 
 * First Basic Measurements
 :PROPERTIES:
@@ -305,18 +335,19 @@ This may be due to the fact that the manufacturer measures the capacitance with
 
 In order to verify that the stroke of the APA300ML is as specified in the datasheet, one side of the APA is fixed to the granite, and a displacement probe[fn:2] 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.
+Then, the voltage across the two actuator stacks is varied from $-20\,V$ to $150\,V$ using a DAC[fn:12] and a voltage amplifier[fn:13].
 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_voltage).
 
 #+name: fig:test_apa_stroke_bench
-#+caption: Bench to measured the APA stroke
+#+caption: Bench to measure the APA stroke
 #+attr_latex: :width 0.7\linewidth
 [[file:figs/test_apa_stroke_bench.jpg]]
 
 The measured APA displacement is shown as a function of the applied voltage in Figure ref:fig:test_apa_stroke_hysteresis.
 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.
+The measured stroke is approximately $250\,\mu m$ when using only two of the three stacks.
 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).
+For the NASS, this stroke is sufficient as the positioning errors to be corrected using the nano-hexapod are expected to be in the order of $10\,\mu m$.
 
 It is clear from Figure ref:fig:test_apa_stroke_hysteresis that "APA 3" has an issue compared to the other units.
 This confirms the abnormal electrical measurements made in Section ref:ssec:test_apa_electrical_measurements.
@@ -410,7 +441,7 @@ The flexible modes for the same condition (i.e. one mechanical interface of the
 #+end_figure
 
 #+name: fig:test_apa_meas_setup_modes
-#+caption: Experimental setup to measured flexible modes of the APA300ML. For the bending in the $X$ direction (\subref{fig:test_apa_meas_setup_X_bending}), the impact point is located at the back of the top measurement point. For the bending in the $Y$ direction (\subref{fig:test_apa_meas_setup_Y_bending}), the impact point is located on the back surface of the top interface (on the back of the 2 measurements points).
+#+caption: Experimental setup to measure flexible modes of the APA300ML. For the bending in the $X$ direction (\subref{fig:test_apa_meas_setup_X_bending}), the impact point is located at the back of the top measurement point. For the bending in the $Y$ direction (\subref{fig:test_apa_meas_setup_Y_bending}), the impact point is located on the back surface of the top interface (on the back of the 2 measurements points).
 #+attr_latex: :options [htbp]
 #+begin_figure
 #+attr_latex: :caption \subcaption{\label{fig:test_apa_meas_setup_X_bending}$X$ bending}
@@ -451,7 +482,7 @@ bending_Y = load('apa300ml_bending_Y_top.mat');
 
 The measured frequency response functions computed from the experimental setups of figures ref:fig:test_apa_meas_setup_X_bending and ref:fig:test_apa_meas_setup_Y_bending are shown in Figure ref:fig:test_apa_meas_freq_compare.
 The $y$ bending mode is observed at $280\,\text{Hz}$ and the $x$ bending mode is at $412\,\text{Hz}$.
-These modes are measured at higher frequencies than the estimated frequencies from the Finite Element Model (see frequencies in Figure ref:fig:test_apa_meas_setup_modes).
+These modes are measured at higher frequencies than the estimated frequencies from the Finite Element Model (see frequencies in Figure ref:fig:test_apa_mode_shapes).
 This is opposite to what is usually observed (i.e. having lower resonance frequencies in practice than the estimation from a finite element model).
 This could be explained by underestimation of the Young's modulus of the steel used for the shell (190 GPa was used for the model, but steel with Young's modulus of 210 GPa could have been used).
 Another explanation is the shape difference between the manufactured APA300ML and the 3D model, for instance thicker blades.
@@ -475,7 +506,7 @@ exportFig('figs/test_apa_meas_freq_compare.pdf', 'width', 'wide', 'height', 'nor
 #+end_src
 
 #+name: fig:test_apa_meas_freq_compare
-#+caption: Obtained frequency response functions for the 2 tests with the instrumented hammer and the laser vibrometer. The Y-bending mode is measured at $280\,\text{Hz}$ and the X-bending mode at $412\,\text{Hz}$
+#+caption: Frequency response functions for the 2 tests with the instrumented hammer and the laser vibrometer. The Y-bending mode is measured at $280\,\text{Hz}$ and the X-bending mode at $412\,\text{Hz}$
 #+RESULTS:
 [[file:figs/test_apa_meas_freq_compare.png]]
 
@@ -485,7 +516,7 @@ exportFig('figs/test_apa_meas_freq_compare.pdf', 'width', 'wide', 'height', 'nor
 :END:
 <<sec:test_apa_dynamics>>
 ** Introduction                                                      :ignore:
-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 dynamics of the APA300ML.
+After the measurements on the APA were performed in Section ref:sec:test_apa_basic_meas, a new test bench is used to better characterize the dynamics of the APA300ML.
 This test bench, depicted in Figure ref:fig:test_bench_apa, comprises the APA300ML fixed at one end to a stationary granite block, and at the other end to a 5kg granite block that is vertically guided by an air bearing.
 That way, there is no friction when actuating the APA300ML, and it will be easier to characterize its behavior independently of other factors.
 An encoder[fn:8] is utilized to measure the relative movement between the two granite blocks, thereby measuring the axial displacement of the APA.
@@ -516,7 +547,7 @@ In order to limit the low frequency gain of the transfer function from $u$ to $V
 Finally, the Integral Force Feedback is implemented, and the amount of damping added is experimentally estimated in Section ref:ssec:test_apa_iff_locus.
 
 #+name: fig:test_apa_schematic
-#+caption: Schematic of the Test Bench used to measured the dynamics of the APA300ML. $u$ is the output DAC voltage, $V_a$ the output amplifier voltage (i.e. voltage applied across the actuator stacks), $d_e$ the measured displacement by the encoder and $V_s$ the measured voltage across the sensor stack.
+#+caption: Schematic of the Test Bench used to measure the dynamics of the APA300ML. $u$ is the output DAC voltage, $V_a$ the output amplifier voltage (i.e. voltage applied across the actuator stacks), $d_e$ the measured displacement by the encoder and $V_s$ the measured voltage across the sensor stack.
 #+attr_latex: :scale 1
 [[file:figs/test_apa_schematic.png]]
 
@@ -581,15 +612,15 @@ exportFig('figs/test_apa_meas_hysteresis.pdf', 'width', 'wide', 'height', 'norma
 #+end_src
 
 #+name: fig:test_apa_meas_hysteresis
-#+caption: Obtained hysteresis curves (displacement as a function of applied voltage) for multiple excitation amplitudes
+#+caption: Displacement as a function of applied voltage for multiple excitation amplitudes
 #+RESULTS:
 [[file:figs/test_apa_meas_hysteresis.png]]
 
 ** Axial stiffness
 <<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.
+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 $\Delta d_e$ is measured using the encoder.
-The APA stiffness can then be estimated from equation eqref:eq:test_apa_stiffness.
+The APA stiffness can then be estimated from equation eqref:eq:test_apa_stiffness, with $g \approx 9.8\,m/s^2$ the acceleration of gravity.
 
 \begin{equation} \label{eq:test_apa_stiffness}
   k_{\text{apa}} = \frac{m_a g}{\Delta d_e}
@@ -679,7 +710,7 @@ The stiffness can also be computed using equation eqref:eq:test_apa_res_freq by
 \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.
+The obtained stiffness is $k \approx 2\,N/\mu m$ which is close to the values found in the documentation and by the "static deflection" method.
 
 
 It is important to note that changes to the electrical impedance connected to the piezoelectric stacks impacts the mechanical compliance (or stiffness) of the piezoelectric stack [[cite:&reza06_piezoel_trans_vibrat_contr_dampin chap. 2]].
@@ -688,7 +719,7 @@ To estimate this effect for the APA300ML, its stiffness is estimated using the "
 - $k_{\text{os}}$: piezoelectric stacks left unconnected (or connect to the high impedance ADC)
 - $k_{\text{sc}}$: piezoelectric stacks short circuited (or connected to the voltage amplifier with small output impedance)
 
-It is found that the open-circuit stiffness is estimated at $k_{\text{oc}} \approx 2.3\,N/\mu m$ while the the closed-circuit stiffness $k_{\text{sc}} \approx 1.7\,N/\mu m$.
+It is found that the open-circuit stiffness is estimated at $k_{\text{oc}} \approx 2.3\,N/\mu m$ while the closed-circuit stiffness $k_{\text{sc}} \approx 1.7\,N/\mu m$.
 
 #+begin_src matlab
 %% Load Data
@@ -762,7 +793,7 @@ The obtained frequency response functions are similar to that of a (second order
   The minus sign comes from the fact that an increase in voltage stretches the piezoelectric stack which reduces the height of the APA
 - A lightly damped resonance at $95\,\text{Hz}$
 - A "mass line" up to $\approx 800\,\text{Hz}$, above which additional resonances appear. These additional resonances might be coming from the limited stiffness of the encoder support or from the limited compliance of the APA support.
-  Flexible modes studied in section ref:ssec:test_apa_spurious_resonances seems not to impact the measured axial motion of the actuator.
+  Flexible modes studied in section ref:ssec:test_apa_spurious_resonances seem not to impact the measured axial motion of the actuator.
 
 The dynamics from $u$ to the measured voltage across the sensor stack $V_s$ for the six APA300ML are compared in Figure ref:fig:test_apa_frf_force.
 
@@ -1018,7 +1049,7 @@ ylabel('Amplitude [V/V]'); set(gca, 'XTickLabel',[]);
 hold off;
 ylim([2e-1, 1e0]);
 yticks([0.2, 0.5, 1]);
-legend('location', 'southeast')
+legend('location', 'southeast', 'FontSize', 8);
 
 ax2 = nexttile;
 hold on;
@@ -1117,7 +1148,7 @@ exportFig('figs/test_apa_iff_plant_comp_manual_fit.pdf', 'width', 'wide', 'heigh
 #+end_src
 
 #+name: fig:test_apa_iff_plant_comp_manual_fit
-#+caption: 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)
+#+caption: 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). Note that a minimum-phase zero is here identified even though the coherence is not good arround the frequency of the zero.
 #+RESULTS:
 [[file:figs/test_apa_iff_plant_comp_manual_fit.png]]
 
@@ -1125,7 +1156,7 @@ The implemented Integral Force Feedback Controller transfer function is shown in
 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}
+K_{\textsc{iff}}(s) = -10 \cdot g \cdot \frac{s}{s + 2\pi \cdot 2} \cdot \frac{1}{s + 2\pi \cdot 20} \cdot \frac{1}{s + 2\pi\cdot 2000}
 \end{equation}
 
 #+begin_src matlab
@@ -1219,7 +1250,7 @@ for i = 1:length(i_kept)
 end
 hold off;
 set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
-xlabel('Frequency [Hz]'); ylabel('Amplitude $d_L/V_a$ [m/V]');
+xlabel('Frequency [Hz]'); ylabel('Amplitude $d_e/u^\prime$ [m/V]');
 xlim([10, 1e3]);
 legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
 #+end_src
@@ -1259,7 +1290,7 @@ ylabel('Imaginary Part')
 axis equal
 ylim([0, 610]);
 xlim([-300,0]);
-legend('location', 'southwest');
+legend('location', 'southwest', 'FontSize', 8);
 #+end_src
 
 #+begin_src matlab :tangle no :exports results :results file none
@@ -1267,7 +1298,7 @@ exportFig('figs/test_apa_iff_root_locus.pdf', 'width', 'half', 'height', 'tall')
 #+end_src
 
 #+name: fig:test_apa_iff
-#+caption: Experimental results of applying Integral Force Feedback to the APA300ML. Obtained damped plant (\subref{fig:test_apa_identified_damped_plants}) and Root Locus (\subref{fig:test_apa_iff_root_locus})
+#+caption: Experimental results of applying Integral Force Feedback to the APA300ML. Obtained damped plant (\subref{fig:test_apa_identified_damped_plants}) and Root Locus (\subref{fig:test_apa_iff_root_locus}) corresponding to the implemented IFF controller \eqref{eq:test_apa_Kiff_formula}
 #+attr_latex: :options [htbp]
 #+begin_figure
 #+attr_latex: :caption \subcaption{\label{fig:test_apa_identified_damped_plants}Measured frequency response functions of damped plants for several IFF gains (solid lines). Identified 2nd order plants to match the experimental data (dashed lines)}
@@ -1291,19 +1322,18 @@ exportFig('figs/test_apa_iff_root_locus.pdf', 'width', 'half', 'height', 'tall')
 :END:
 <<sec:test_apa_model_2dof>>
 
-** Introduction                                                      :ignore:
+**** Introduction                                                  :ignore:
-In this section, a simscape model (Figure ref:fig:test_apa_bench_model) of the measurement bench is used to compare the model of the APA with the measured frequency response functions.
+In this section, a Simscape model (Figure ref:fig:test_apa_bench_model) of the measurement bench is used to tune the 2 degrees of freedom model of the APA using the measured frequency response functions.
 
-A 2 degrees of freedom model is used to model the APA300ML.
+This 2 degrees of freedom model is developed to accurately represent the APA300ML dynamics while having low complexity and low number of associated states.
-This model is presented in Section ref:ssec:test_apa_2dof_model and the procedure to tuned the model is described in Section ref:ssec:test_apa_2dof_model_tuning.
+After the model presented, the procedure to tune the model is described and the obtained model dynamics is compared with the measurements.
-The obtained model dynamics is compared with the measurements in Section ref:ssec:test_apa_2dof_model_result.
 
 #+name: fig:test_apa_bench_model
 #+caption: Screenshot of the Simscape model
 #+attr_latex: :width 0.8\linewidth
 [[file:figs/test_apa_bench_model.png]]
 
-** Matlab Init                                              :noexport:ignore:
+**** Matlab Init                                          :noexport:ignore:
 #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
 <<matlab-dir>>
 #+end_src
@@ -1321,11 +1351,11 @@ The obtained model dynamics is compared with the measurements in Section ref:sse
 #+end_src
 
 #+begin_src matlab :tangle no :noweb yes
-<<m-init-path-simscape>>
+<<m-init-path-Simscape>>
 #+end_src
 
 #+begin_src matlab :eval no :noweb yes
-<<m-init-path-simscape-tangle>>
+<<m-init-path-Simscape-tangle>>
 #+end_src
 
 #+begin_src matlab :noweb yes
@@ -1343,17 +1373,16 @@ io(io_i) = linio([mdl, '/de'], 1, 'openoutput'); io_i = io_i + 1; % Encoder
 freqs = 5*logspace(0, 3, 1000);
 #+end_src
 
-** Two Degrees of Freedom APA Model
+**** Two Degrees of Freedom APA Model
-<<ssec:test_apa_2dof_model>>
 
 The model of the amplified piezoelectric actuator is shown in Figure ref:fig:test_apa_2dof_model.
 It can be decomposed into three components:
 - the shell whose axial properties are represented by $k_1$ and $c_1$
 - the actuator stacks whose contribution in the axial stiffness is represented by $k_a$ and $c_a$.
   A force source $\tau$ represents the axial force induced by the force sensor stacks.
-  The gain $g_a$ (in $N/m$) is used to convert the applied voltage $V_a$ to the axial force $\tau$
+  The sensitivity $g_a$ (in $N/m$) is used to convert the applied voltage $V_a$ to the axial force $\tau$
 - the sensor stack whose contribution in the axial stiffness is represented by $k_e$ and $c_e$.
-  A  sensor measures the stack strain $d_L$ which is then converted to a voltage $V_s$ using a gain $g_s$ (in $V/m$)
+  A sensor measures the stack strain $d_e$ which is then converted to a voltage $V_s$ using a sensitivity $g_s$ (in $V/m$)
 
 Such simple model has some limitations:
 - it only represents the axial characteristics of the APA as it is modelled as infinitely rigid in the other directions
@@ -1364,14 +1393,13 @@ Such simple model has some limitations:
 #+caption: Schematic of the two degrees of freedom model of the APA300ML, adapted from cite:souleille18_concep_activ_mount_space_applic
 [[file:figs/test_apa_2dof_model.png]]
 
-** Tuning of the APA model
+**** Tuning of the APA model                                       :ignore:
-<<ssec:test_apa_2dof_model_tuning>>
 
-9 parameters ($m$, $k_1$, $c_1$, $k_e$, $c_e$, $k_a$, $c_a$, $g_s$ and $g_a$) have to be tuned such that the dynamics of the model (Figure ref:fig:test_apa_2dof_model_simscape) well represents the identified dynamics in Section ref:sec:test_apa_dynamics.
+9 parameters ($m$, $k_1$, $c_1$, $k_e$, $c_e$, $k_a$, $c_a$, $g_s$ and $g_a$) have to be tuned such that the dynamics of the model (Figure ref:fig:test_apa_2dof_model_Simscape) well represents the identified dynamics in Section ref:sec:test_apa_dynamics.
 
-#+name: fig:test_apa_2dof_model_simscape
+#+name: fig:test_apa_2dof_model_Simscape
 #+caption: Schematic of the two degrees of freedom model of the APA300ML with input $V_a$ and outputs $d_e$ and $V_s$
-[[file:figs/test_apa_2dof_model_simscape.png]]
+[[file:figs/test_apa_2dof_model_Simscape.png]]
 
 #+begin_src matlab
 %% Stiffness values for the 2DoF APA model
@@ -1391,8 +1419,8 @@ ce = 2*ca; % Damping of the sensor stack [N/(m/s)]
 #+end_src
 
 #+begin_src matlab
-%% Estimation ot the sensor and actuator gains
+%% Estimation ot the sensor and actuator sensitivities
-% Initialize the structure with unitary sensor and actuator "gains"
+% Initialize the structure with unitary sensor and actuator "sensitivities"
 n_hexapod = struct();
 n_hexapod.actuator = initializeAPA(...
     'type', '2dof', ...
@@ -1413,15 +1441,15 @@ G_norm = linearize(mdl, io, 0.0, opts);
 G_norm.InputName  = {'u'};
 G_norm.OutputName = {'Vs', 'de'};
 
-% Load Identification Data to estimate the two gains
+% Load Identification Data to estimate the two sensitivities
 load('meas_apa_frf.mat', 'f', 'Ts', 'enc_frf', 'iff_frf', 'apa_nums');
 
-% Estimation ot the Actuator Gain
+% Estimation ot the Actuator sensitivity
 fa = 10; % Frequency where the two FRF should match [Hz]
 [~, i_f] = min(abs(f - fa));
 ga = -abs(enc_frf(i_f,1))./abs(evalfr(G_norm('de', 'u'), 1i*2*pi*fa));
 
-% Estimation ot the Sensor Gain
+% Estimation ot the Sensor sensitivity
 fs = 600; % Frequency where the two FRF should match [Hz]
 [~, i_f] = min(abs(f - fs));
 gs = -abs(iff_frf(i_f,1))./abs(evalfr(G_norm('Vs', 'u'), 1i*2*pi*fs))/ga;
@@ -1450,9 +1478,9 @@ Knowing from eqref:eq:test_apa_tot_stiffness that the total stiffness is $k_{\te
 Then, $c_a$ (and therefore $c_e = 2 c_a$) can be tuned to match the damping ratio of the identified resonance.
 $c_a = 100\,Ns/m$ and $c_e = 200\,Ns/m$ are obtained.
 
-Finally, the two gains $g_s$ and $g_a$ can be tuned to match the gain of the identified transfer functions.
+Finally $g_s$ and $g_a$ can be tuned to match the gain of the identified transfer functions.
 
-The obtained parameters of the model shown in Figure ref:fig:test_apa_2dof_model_simscape are summarized in Table ref:tab:test_apa_2dof_parameters.
+The obtained parameters of the model shown in Figure ref:fig:test_apa_2dof_model_Simscape are summarized in Table ref:tab:test_apa_2dof_parameters.
 
 #+name: tab:test_apa_2dof_parameters
 #+caption: Summary of the obtained parameters for the 2 DoF APA300ML model
@@ -1470,8 +1498,7 @@ The obtained parameters of the model shown in Figure ref:fig:test_apa_2dof_model
 | $g_a$       | $-2.58\,N/V$     |
 | $g_s$       | $0.46\,V/\mu m$  |
 
-** Obtained Dynamics
+**** Obtained Dynamics                                             :ignore:
-<<ssec:test_apa_2dof_model_result>>
 
 The dynamics of the two degrees of freedom model of the APA300ML is now extracted using optimized parameters (listed in Table ref:tab:test_apa_2dof_parameters) from the Simscape model.
 It is compared with the experimental data in Figure ref:fig:test_apa_2dof_comp_frf.
@@ -1601,10 +1628,10 @@ exportFig('figs/test_apa_2dof_comp_frf_force.pdf', 'width', 'half', 'height', 't
 :END:
 <<sec:test_apa_model_flexible>>
 
-** Introduction                                                      :ignore:
+**** Introduction                                                  :ignore:
 In this section, a /super element/ of the APA300ML is computed using a finite element software[fn:11].
 It is then imported in Simscape (in the form of a stiffness matrix and a mass matrix) and included in the same model that was used in ref:sec:test_apa_model_2dof.
-This procedure is illustrated in Figure ref:fig:test_apa_super_element_simscape.
+This procedure is illustrated in Figure ref:fig:test_apa_super_element_Simscape.
 Several /remote points/ are defined in the finite element model (here illustrated by colorful planes and numbers from =1= to =5=) and are then make accessible in the Simscape model as shown at the right by the "frames" =F1= to =F5=.
 
 For the APA300ML /super element/, 5 /remote points/ are defined.
@@ -1613,12 +1640,12 @@ Two /remote points/ (=3= and =4=) are located across two piezoelectric stacks an
 Finally, two /remote points/ (=4= and =4=) are located across the third piezoelectric stack.
 It will be used to measure the strain experience by this stack, and model the sensor stack.
 
-#+name: fig:test_apa_super_element_simscape
+#+name: fig:test_apa_super_element_Simscape
 #+attr_latex: :width 1.0\linewidth
 #+caption: Finite Element Model of the APA300ML with "remotes points" on the left. Simscape model with included "Reduced Order Flexible Solid" on the right.
-[[file:figs/test_apa_super_element_simscape.png]]
+[[file:figs/test_apa_super_element_Simscape.png]]
 
-** Matlab Init                                              :noexport:ignore:
+**** Matlab Init                                          :noexport:ignore:
 #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
 <<matlab-dir>>
 #+end_src
@@ -1636,11 +1663,11 @@ It will be used to measure the strain experience by this stack, and model the se
 #+end_src
 
 #+begin_src matlab :tangle no :noweb yes
-<<m-init-path-simscape>>
+<<m-init-path-Simscape>>
 #+end_src
 
 #+begin_src matlab :eval no :noweb yes
-<<m-init-path-simscape-tangle>>
+<<m-init-path-Simscape-tangle>>
 #+end_src
 
 #+begin_src matlab :noweb yes
@@ -1658,8 +1685,7 @@ io(io_i) = linio([mdl, '/de'], 1, 'openoutput'); io_i = io_i + 1; % Encoder
 freqs = 5*logspace(0, 3, 1000);
 #+end_src
 
-** Identification of the Actuator and Sensor constants
+**** Identification of the Actuator and Sensor constants
-<<ssec:test_apa_flexible_ga_gs>>
 
 Once the APA300ML /super element/ is included in the Simscape model, the transfer function from $F_a$ to $d_L$ and $d_e$ can be extracted.
 The gains $g_a$ and $g_s$ are then be tuned such that the gain of the transfer functions are matching the identified ones.
@@ -1690,7 +1716,7 @@ ga = -mean(abs(enc_frf(f>10 & f<20)))./dcgain(G_norm('de', 'u'));
 gs = -mean(abs(iff_frf(f>400 & f<500)))./(ga*abs(squeeze(freqresp(G_norm('Vs', 'u'), 1e3, 'Hz'))));
 #+end_src
 
-To make sure these "gains" are physically valid, it is possible to estimate them from physical properties of the piezoelectric stack material.
+To make sure the sensitivities $g_a$ and $g_s$ are physically valid, it is possible to estimate them from physical properties of the piezoelectric stack material.
 
 From [[cite:&fleming14_desig_model_contr_nanop_system p. 123]], the relation between relative displacement $d_L$ of the sensor stack and generated voltage $V_s$ is given by eqref:eq:test_apa_piezo_strain_to_voltage and from [[cite:&fleming10_integ_strain_force_feedb_high]] the relation between the force $F_a$ and the applied voltage $V_a$ is given by eqref:eq:test_apa_piezo_voltage_to_force.
 
@@ -1708,7 +1734,7 @@ The properties of this "THP5H" material used to compute $g_a$ and $g_s$ are list
 From these parameters, $g_s = 5.1\,V/\mu m$ and $g_a = 26\,N/V$ were obtained which are close to the identified constants using the experimentally identified transfer functions.
 
 #+name: tab:test_apa_piezo_properties
-#+caption: Piezoelectric properties used for the estimation of the sensor and actuators "gains"
+#+caption: Piezoelectric properties used for the estimation of the sensor and actuators sensitivities
 #+attr_latex: :environment tabularx :width 1\linewidth :align ccX
 #+attr_latex: :center t :booktabs t
 | *Parameter*    | *Value*                    | *Description*                                                |
@@ -1742,12 +1768,11 @@ ka  = cE*A/L;    % Stiffness of the two stacks [N/m]
 ga_th = d33*n*ka;   % Actuator Constant [N/V]
 #+end_src
 
-** Comparison of the obtained dynamics
+**** Comparison of the obtained dynamics
-<<ssec:test_apa_flexible_comp_frf>>
 
-The obtained dynamics using the /super element/ with the tuned "sensor gain" and "actuator gain" are compared with the experimentally identified frequency response functions in Figure ref:fig:test_apa_super_element_comp_frf.
+The obtained dynamics using the /super element/ with the tuned "sensor sensitivity" and "actuator sensitivity" are compared with the experimentally identified frequency response functions in Figure ref:fig:test_apa_super_element_comp_frf.
 A good match between the model and the experimental results is observed.
-It is however a bit surprising that the model is a bit "softer" than the measured system as finite element models are usually overestimating the stiffness.
+It is however surprising that the model is "softer" than the measured system as finite element models are usually overestimating the stiffness (see Section ref:ssec:test_apa_spurious_resonances for possible explanations).
 
 Using this simple test bench, it can be concluded that the /super element/ model of the APA300ML well captures the axial dynamics of the actuator (the actuator stacks, the force sensor stack as well as the shell used as a mechanical lever).
 
@@ -1756,8 +1781,8 @@ Using this simple test bench, it can be concluded that the /super element/ model
 % Initialize the APA
 n_hexapod.actuator = initializeAPA(...
     'type', 'flexible', ...
-    'ga',   23.2, ... % Actuator gain [N/V]
+    'ga',   23.2, ... % Actuator sensitivity [N/V]
-    'gs',   -4.9e6);    % Sensor gain [V/m]
+    'gs',   -4.9e6);    % Sensor sensitivity [V/m]
 
 % Identify with updated constants
 G_flex = exp(-Ts*s)*linearize(mdl, io, 0.0, opts);
@@ -1844,7 +1869,7 @@ exportFig('figs/test_apa_super_element_comp_frf_force.pdf', 'width', 'half', 'he
 #+end_src
 
 #+name: fig:test_apa_super_element_comp_frf
-#+caption: Comparison of the measured frequency response functions and the identified dynamics from the "flexible" model of the APA300ML. Both for the dynamics from $u$ to $d_e$ (\subref{fig:test_apa_2dof_comp_frf_enc}) (\subref{fig:test_apa_2dof_comp_frf_force}) and from $u$ to $V_s$ (\subref{fig:test_apa_2dof_comp_frf_force})
+#+caption: Comparison of the measured frequency response functions and the identified dynamics from the finite element model of the APA300ML. Both for the dynamics from $u$ to $d_e$ (\subref{fig:test_apa_2dof_comp_frf_enc}) (\subref{fig:test_apa_2dof_comp_frf_force}) and from $u$ to $V_s$ (\subref{fig:test_apa_2dof_comp_frf_force})
 #+attr_latex: :options [htbp]
 #+begin_figure
 #+attr_latex: :caption \subcaption{\label{fig:test_apa_super_element_comp_frf_enc}from $u$ to $d_e$}
@@ -1864,7 +1889,7 @@ exportFig('figs/test_apa_super_element_comp_frf_force.pdf', 'width', 'half', 'he
 * Conclusion
 <<sec:test_apa_conclusion>>
 
-In this study, the received amplified piezoelectric actuators "APA300ML" have been characterized to make sure they are fulfilling all the requirements determined during the detailed design phase.
+In this study, the amplified piezoelectric actuators "APA300ML" have been characterized to make sure they are fulfilling all the requirements determined during the detailed design phase.
 
 Geometrical features such as the flatness of its interfaces, electrical capacitance and achievable strokes were measured in Section ref:sec:test_apa_basic_meas.
 These simple measurements allowed for early detection of a manufacturing defect in one of the APA300ML.
@@ -1875,7 +1900,7 @@ Although a non-minimum zero was identified in the transfer function from $u$ to
 
 Then, two different models were used to represent the APA300ML dynamics.
 In Section ref:sec:test_apa_model_2dof, a simple two degrees of freedom mass-spring-damper model was presented and tuned based on the measured dynamics.
-After following a tuning procedure (see Section ref:ssec:test_apa_2dof_model_tuning), the model dynamics was shown to match very well with the experiment.
+After following a tuning procedure, the model dynamics was shown to match very well with the experiment.
 However, it is important to note that this model only represents the axial dynamics of the actuators, assuming infinite stiffness in other directions.
 
 In Section ref:sec:test_apa_model_flexible, a /super element/ extracted from a finite element model was used to model the APA300ML.
@@ -1905,7 +1930,7 @@ addpath('./mat/'); % Path for data
 #+END_SRC
 
 ** Initialize Simscape
-#+NAME: m-init-path-simscape
+#+NAME: m-init-path-Simscape
 #+BEGIN_SRC matlab
 addpath('./matlab/STEPS/'); % Path for Simscape Model
 
@@ -1914,11 +1939,11 @@ opts = linearizeOptions;
 opts.SampleTime = 0;
 
 %% Open Simscape Model
-mdl = 'test_apa_simscape'; % Name of the Simulink File
+mdl = 'test_apa_Simscape'; % Name of the Simulink File
 open(mdl); % Open Simscape Model
 #+END_SRC
 
-#+NAME: m-init-path-simscape-tangle
+#+NAME: m-init-path-Simscape-tangle
 #+BEGIN_SRC matlab
 addpath('./STEPS/'); % Path for Simscape Model
 
@@ -1927,7 +1952,7 @@ opts = linearizeOptions;
 opts.SampleTime = 0;
 
 %% Open Simscape Model
-mdl = 'test_apa_simscape'; % Name of the Simulink File
+mdl = 'test_apa_Simscape'; % Name of the Simulink File
 open(mdl); % Open Simscape Model
 #+END_SRC
 
@@ -2038,7 +2063,7 @@ if args.Ga == 0
         actuator.Ga = 23.2;
     end
 else
-    actuator.Ga = args.Ga; % Actuator gain [N/V]
+    actuator.Ga = args.Ga; % Actuator sensitivity [N/V]
 end
 #+end_src
 
@@ -2051,7 +2076,7 @@ if args.Gs == 0
         actuator.Gs = -4898341;
     end
 else
-    actuator.Gs = args.Gs; % Sensor gain [V/m]
+    actuator.Gs = args.Gs; % Sensor sensitivity [V/m]
 end
 #+end_src
 
@@ -2100,6 +2125,8 @@ actuator.cs = args.cs; % Damping of one stack [N/m]
 
 * Footnotes
 
+[fn:13]PD200 from PiezoDrive. The gain is $20\,V/V$
+[fn:12]The DAC used is the one included in the IO133 card sold by Speedgoat. It has an output range of $\pm 10\,V$ and 16-bits resolution
 [fn:11]Ansys\textsuperscript{\textregistered} was used
 [fn:10]The transfer function fitting was computed using the =vectfit3= routine, see  [[cite:&gustavsen99_ration_approx_frequen_domain_respon]]
 [fn:9]Frequency of the sinusoidal wave is $1\,\text{Hz}$

test-bench-apa.tex

@@ -1,7 +1,26 @@
-% Created 2024-04-04 Thu 11:14
+% Created 2024-04-30 Tue 16:37
 % Intended LaTeX compiler: pdflatex
 \documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
 
+\newacronym{haclac}{HAC-LAC}{High Authority Control - Low Authority Control}
+\newacronym{hac}{HAC}{High Authority Control}
+\newacronym{lac}{LAC}{Low Authority Control}
+\newacronym{nass}{NASS}{Nano Active Stabilization System}
+\newacronym{asd}{ASD}{Amplitude Spectral Density}
+\newacronym{psd}{PSD}{Power Spectral Density}
+\newacronym{cps}{CPS}{Cumulative Power Spectrum}
+\newacronym{cas}{CAS}{Cumulative Amplitude Spectrum}
+\newacronym{frf}{FRF}{Frequency Response Function}
+\newacronym{iff}{IFF}{Integral Force Feedback}
+\newacronym{rdc}{RDC}{Relative Damping Control}
+\newacronym{drga}{DRGA}{Dynamical Relative Gain Array}
+\newacronym{hpf}{HPF}{high-pass filter}
+\newacronym{lpf}{LPF}{low-pass filter}
+\newacronym{dof}{DoF}{Degrees of Freedom}
+\newglossaryentry{psdx}{name=\ensuremath{\Phi_{x}},description={{Power spectral density of signal $x$}}}
+\newglossaryentry{asdx}{name=\ensuremath{\Gamma_{x}},description={{Amplitude spectral density of signal $x$}}}
+\newglossaryentry{cpsx}{name=\ensuremath{\Phi_{x}},description={{Cumulative Power Spectrum of signal $x$}}}
+\newglossaryentry{casx}{name=\ensuremath{\Gamma_{x}},description={{Cumulative Amplitude Spectrum of signal $x$}}}
 \input{preamble.tex}
 \bibliography{test-bench-apa.bib}
 \author{Dehaeze Thomas}
@@ -12,7 +31,7 @@
  pdftitle={Test Bench - Amplified Piezoelectric Actuator},
  pdfkeywords={},
  pdfsubject={},
- pdfcreator={Emacs 29.3 (Org mode 9.7)}, 
+ pdfcreator={Emacs 29.3 (Org mode 9.6)}, 
  pdflang={English}}
 \usepackage{biblatex}
 
@@ -22,15 +41,11 @@
 \tableofcontents
 
 \clearpage
+
 In this chapter, the goal is to make sure that the received APA300ML (shown in Figure \ref{fig:test_apa_received}) are complying with the requirements and that dynamical models of the actuator are well representing its dynamics.
 
-\begin{figure}[htbp]
+In section \ref{sec:test_apa_basic_meas}, the mechanical tolerances of the APA300ML interfaces are checked together with the electrical properties of the piezoelectric stacks, the achievable stroke.
-\centering
+Flexible modes of the APA300ML which were estimated using a finite element model are compared with measurements.
-\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}
-
-In section \ref{sec:test_apa_basic_meas}, the mechanical tolerances of the APA300ML interfaces are checked together with the electrical properties of the piezoelectric stacks, the achievable stroke. Flexible modes of the APA300ML are computed with a finite element model and compared with measurements.
 
 Using a dedicated test bench, dynamical measurements are performed (Section \ref{sec:test_apa_dynamics}).
 The dynamics from the generated DAC voltage (going through the voltage amplifier and then to two actuator stacks) to the induced axial displacement and to the measured voltage across the force sensor stack are estimated.
@@ -38,37 +53,27 @@ Integral Force Feedback is experimentally applied and the damped plants are esti
 
 Two different models of the APA300ML are then presented.
 First, in Section \ref{sec:test_apa_model_2dof}, a two degrees of freedom model is presented, tuned and compared with the measured dynamics.
-This model is proven to accurately simulate the APA300ML's axial dynamics.
+This model is proven to accurately represents the APA300ML's axial dynamics while having low complexity.
 
 Then, in Section \ref{sec:test_apa_model_flexible}, a \emph{super element} of the APA300ML is extracted using a finite element model and imported in Simscape.
 This more complex model is also shown to well capture the axial dynamics of the APA300ML.
 
-\begin{table}[htbp]
+\begin{figure}[htbp]
 \centering
-\begin{tabularx}{0.6\linewidth}{lX}
+\includegraphics[scale=1,width=0.7\linewidth]{figs/test_apa_received.jpg}
-\toprule
+\caption{\label{fig:test_apa_received}Picture of 5 out of the 7 received APA300ML}
-\textbf{Sections} & \textbf{Matlab File}\\
+\end{figure}
-\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\_dynamics.m}\\
-Section \ref{sec:test_apa_model_2dof} & \texttt{test\_apa\_3\_model\_2dof.m}\\
-Section \ref{sec:test_apa_model_flexible} & \texttt{test\_apa\_4\_model\_flexible.m}\\
-\bottomrule
-\end{tabularx}
-\caption{\label{tab:test_apa_section_matlab_code}Report sections and corresponding Matlab files}
 
-\end{table}
 \chapter{First Basic Measurements}
-\label{sec:org228596b}
 \label{sec:test_apa_basic_meas}
-
 Before measuring the dynamical characteristics of the APA300ML, first simple measurements are performed.
 First, the tolerances (especially flatness) of the mechanical interfaces are checked in Section \ref{ssec:test_apa_geometrical_measurements}.
 Then, the capacitance of the piezoelectric stacks is measured in Section \ref{ssec:test_apa_electrical_measurements}.
 The achievable stroke of the APA300ML is measured using a displacement probe in Section \ref{ssec:test_apa_stroke_measurements}.
 Finally, in Section \ref{ssec:test_apa_spurious_resonances}, the flexible modes of the APA are measured and compared with a finite element model.
+
 \section{Geometrical Measurements}
-\label{sec:org726c24d}
 \label{ssec: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 excellent flatness.
@@ -102,8 +107,8 @@ APA 7 & 18.7\\
 
 \end{center}
 \end{minipage}
+
 \section{Electrical Measurements}
-\label{sec:orgd08d506}
 \label{ssec: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 \(20\,\mu F\).
@@ -143,25 +148,26 @@ APA 7 & 4.85 & 9.85\\
 
 \end{center}
 \end{minipage}
+
 \section{Stroke and Hysteresis Measurement}
-\label{sec:org1a772f2}
 \label{ssec:test_apa_stroke_measurements}
 
 In order to verify that the stroke of the APA300ML is as specified in the datasheet, 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.
+Then, the voltage across the two actuator stacks is varied from \(-20\,V\) to \(150\,V\) using a DAC\footnote{The DAC used is the one included in the IO133 card sold by Speedgoat. It has an output range of \(\pm 10\,V\) and 16-bits resolution} and a voltage amplifier\footnote{PD200 from PiezoDrive. The gain is \(20\,V/V\)}.
 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_voltage}).
 
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1,width=0.7\linewidth]{figs/test_apa_stroke_bench.jpg}
-\caption{\label{fig:test_apa_stroke_bench}Bench to measured the APA stroke}
+\caption{\label{fig:test_apa_stroke_bench}Bench to measure 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_hysteresis}.
 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.
+The measured stroke is approximately \(250\,\mu m\) when using only two of the three stacks.
 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).
+For the NASS, this stroke is sufficient as the positioning errors to be corrected using the nano-hexapod are expected to be in the order of \(10\,\mu m\).
 
 It is clear from Figure \ref{fig:test_apa_stroke_hysteresis} that ``APA 3'' has an issue compared to the other units.
 This confirms the abnormal electrical measurements made in Section \ref{ssec:test_apa_electrical_measurements}.
@@ -183,8 +189,8 @@ From now on, only the six APA that behave as expected will be used.
 \end{subfigure}
 \caption{\label{fig:test_apa_stroke}Generated voltage across the two piezoelectric stack actuators to estimate the stroke of the APA300ML (\subref{fig:test_apa_stroke_voltage}). Measured displacement as a function of the applied voltage (\subref{fig:test_apa_stroke_hysteresis})}
 \end{figure}
+
 \section{Flexible Mode Measurement}
-\label{sec:orgf072115}
 \label{ssec:test_apa_spurious_resonances}
 
 In this section, the flexible modes of the APA300ML are investigated both experimentally and using a Finite Element Model.
@@ -229,12 +235,12 @@ The flexible modes for the same condition (i.e. one mechanical interface of the
 \end{center}
 \subcaption{\label{fig:test_apa_meas_setup_Y_bending}$Y$ Bending}
 \end{subfigure}
-\caption{\label{fig:test_apa_meas_setup_modes}Experimental setup to measured flexible modes of the APA300ML. For the bending in the \(X\) direction (\subref{fig:test_apa_meas_setup_X_bending}), the impact point is located at the back of the top measurement point. For the bending in the \(Y\) direction (\subref{fig:test_apa_meas_setup_Y_bending}), the impact point is located on the back surface of the top interface (on the back of the 2 measurements points).}
+\caption{\label{fig:test_apa_meas_setup_modes}Experimental setup to measure flexible modes of the APA300ML. For the bending in the \(X\) direction (\subref{fig:test_apa_meas_setup_X_bending}), the impact point is located at the back of the top measurement point. For the bending in the \(Y\) direction (\subref{fig:test_apa_meas_setup_Y_bending}), the impact point is located on the back surface of the top interface (on the back of the 2 measurements points).}
 \end{figure}
 
 The measured frequency response functions computed from the experimental setups of figures \ref{fig:test_apa_meas_setup_X_bending} and \ref{fig:test_apa_meas_setup_Y_bending} are shown in Figure \ref{fig:test_apa_meas_freq_compare}.
 The \(y\) bending mode is observed at \(280\,\text{Hz}\) and the \(x\) bending mode is at \(412\,\text{Hz}\).
-These modes are measured at higher frequencies than the estimated frequencies from the Finite Element Model (see frequencies in Figure \ref{fig:test_apa_meas_setup_modes}).
+These modes are measured at higher frequencies than the estimated frequencies from the Finite Element Model (see frequencies in Figure \ref{fig:test_apa_mode_shapes}).
 This is opposite to what is usually observed (i.e. having lower resonance frequencies in practice than the estimation from a finite element model).
 This could be explained by underestimation of the Young's modulus of the steel used for the shell (190 GPa was used for the model, but steel with Young's modulus of 210 GPa could have been used).
 Another explanation is the shape difference between the manufactured APA300ML and the 3D model, for instance thicker blades.
@@ -242,12 +248,12 @@ Another explanation is the shape difference between the manufactured APA300ML an
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1]{figs/test_apa_meas_freq_compare.png}
-\caption{\label{fig:test_apa_meas_freq_compare}Obtained frequency response functions for the 2 tests with the instrumented hammer and the laser vibrometer. The Y-bending mode is measured at \(280\,\text{Hz}\) and the X-bending mode at \(412\,\text{Hz}\)}
+\caption{\label{fig:test_apa_meas_freq_compare}Frequency response functions for the 2 tests with the instrumented hammer and the laser vibrometer. The Y-bending mode is measured at \(280\,\text{Hz}\) and the X-bending mode at \(412\,\text{Hz}\)}
 \end{figure}
+
 \chapter{Dynamical measurements}
-\label{sec:org2e49535}
 \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 dynamics of the APA300ML.
+After the measurements on the APA were performed in Section \ref{sec:test_apa_basic_meas}, a new test bench is used to better characterize the dynamics of the APA300ML.
 This test bench, depicted in Figure \ref{fig:test_bench_apa}, comprises the APA300ML fixed at one end to a stationary granite block, and at the other end to a 5kg granite block that is vertically guided by an air bearing.
 That way, there is no friction when actuating the APA300ML, and it will be easier to characterize its behavior independently of other factors.
 An encoder\footnote{Renishaw Vionic, resolution of \(2.5\,nm\)} is utilized to measure the relative movement between the two granite blocks, thereby measuring the axial displacement of the APA.
@@ -278,10 +284,9 @@ Finally, the Integral Force Feedback is implemented, and the amount of damping a
 \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 used to measured the dynamics of the APA300ML. \(u\) is the output DAC voltage, \(V_a\) the output amplifier voltage (i.e. voltage applied across the actuator stacks), \(d_e\) the measured displacement by the encoder and \(V_s\) the measured voltage across the sensor stack.}
+\caption{\label{fig:test_apa_schematic}Schematic of the Test Bench used to measure the dynamics of the APA300ML. \(u\) is the output DAC voltage, \(V_a\) the output amplifier voltage (i.e. voltage applied across the actuator stacks), \(d_e\) the measured displacement by the encoder and \(V_s\) the measured voltage across the sensor stack.}
 \end{figure}
 \section{Hysteresis}
-\label{sec:org63e3610}
 \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.
@@ -292,14 +297,14 @@ This is the typical behavior expected from a PZT stack actuator where the hyster
 \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}
+\caption{\label{fig:test_apa_meas_hysteresis}Displacement as a function of applied voltage for multiple excitation amplitudes}
 \end{figure}
+
 \section{Axial stiffness}
-\label{sec:org68f94d3}
 \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.
+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 \(\Delta d_e\) is measured using the encoder.
-The APA stiffness can then be estimated from equation \eqref{eq:test_apa_stiffness}.
+The APA stiffness can then be estimated from equation \eqref{eq:test_apa_stiffness}, with \(g \approx 9.8\,m/s^2\) the acceleration of gravity.
 
 \begin{equation} \label{eq:test_apa_stiffness}
   k_{\text{apa}} = \frac{m_a g}{\Delta d_e}
@@ -344,7 +349,7 @@ The stiffness can also be computed using equation \eqref{eq:test_apa_res_freq} b
 \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.
+The obtained stiffness is \(k \approx 2\,N/\mu m\) which is close to the values found in the documentation and by the ``static deflection'' method.
 
 
 It is important to note that changes to 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}.
@@ -355,9 +360,9 @@ To estimate this effect for the APA300ML, its stiffness is estimated using the `
 \item \(k_{\text{sc}}\): piezoelectric stacks short circuited (or connected to the voltage amplifier with small output impedance)
 \end{itemize}
 
-It is found that the open-circuit stiffness is estimated at \(k_{\text{oc}} \approx 2.3\,N/\mu m\) while the the closed-circuit stiffness \(k_{\text{sc}} \approx 1.7\,N/\mu m\).
+It is found that the open-circuit stiffness is estimated at \(k_{\text{oc}} \approx 2.3\,N/\mu m\) while the closed-circuit stiffness \(k_{\text{sc}} \approx 1.7\,N/\mu m\).
+
 \section{Dynamics}
-\label{sec:org7856f12}
 \label{ssec:test_apa_meas_dynamics}
 
 In this section, the dynamics from the excitation voltage \(u\) to the encoder measured displacement \(d_e\) and to the force sensor voltage \(V_s\) is identified.
@@ -369,7 +374,7 @@ The obtained frequency response functions are similar to that of a (second order
 The minus sign comes from the fact that an increase in voltage stretches the piezoelectric stack which 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 additional resonances appear. These additional resonances might be coming from the limited stiffness of the encoder support or from the limited compliance of the APA support.
-Flexible modes studied in section \ref{ssec:test_apa_spurious_resonances} seems not to impact the measured axial motion of the actuator.
+Flexible modes studied in section \ref{ssec:test_apa_spurious_resonances} seem not to impact the measured axial motion of the actuator.
 \end{itemize}
 
 The dynamics from \(u\) to the measured voltage across the sensor stack \(V_s\) for the six APA300ML are compared in Figure \ref{fig:test_apa_frf_force}.
@@ -402,8 +407,8 @@ All the identified dynamics of the six APA300ML (both when looking at the encode
 \end{subfigure}
 \caption{\label{fig:test_apa_frf_dynamics}Measured frequency response function from generated voltage \(u\) to the encoder displacement \(d_e\) (\subref{fig:test_apa_frf_encoder}) and to the force sensor voltage \(V_s\) (\subref{fig:test_apa_frf_force}) for the six APA300ML}
 \end{figure}
+
 \section{Non Minimum Phase Zero?}
-\label{sec:org49578de}
 \label{ssec:test_apa_non_minimum_phase}
 
 It was surprising to observe a non-minimum phase behavior for the zero on the transfer function from \(u\) to \(V_s\) (Figure \ref{fig:test_apa_frf_force}).
@@ -432,8 +437,9 @@ However, this is not so important here as the zero is lightly damped (i.e. very
 \end{subfigure}
 \caption{\label{fig:test_apa_non_minimum_phase}Measurement of the anti-resonance found on the transfer function from \(u\) to \(V_s\). The coherence (\subref{fig:test_apa_non_minimum_phase_coherence}) is quite good around the anti-resonance frequency. The phase (\subref{fig:test_apa_non_minimum_phase_zoom}) shoes a non-minimum phase behavior.}
 \end{figure}
+
+
 \section{Effect of the resistor on the IFF Plant}
-\label{sec:org8b60e04}
 \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 piezoelectric stack (capacitance estimated at \(\approx 5\,\mu F\)).
@@ -448,8 +454,8 @@ It is confirmed that the added resistor as the effect of adding an high pass fil
 \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{sec:orga8f1ff3}
 \label{ssec:test_apa_iff_locus}
 
 In order to implement the Integral Force Feedback strategy, the measured frequency response function from \(u\) to \(V_s\) (Figure \ref{fig:test_apa_frf_force}) is fitted using the transfer function shown in equation \eqref{eq:test_apa_iff_manual_fit}.
@@ -464,14 +470,14 @@ The comparison between the identified plant and the manually tuned transfer func
 \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)}
+\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). Note that a minimum-phase zero is here identified even though the coherence is not good arround the frequency of the zero.}
 \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}
+K_{\textsc{iff}}(s) = -10 \cdot g \cdot \frac{s}{s + 2\pi \cdot 2} \cdot \frac{1}{s + 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.
@@ -503,26 +509,24 @@ The two obtained root loci are compared in Figure \ref{fig:test_apa_iff_root_loc
 \end{center}
 \subcaption{\label{fig:test_apa_iff_root_locus}Root Locus plot using the plant model (black) and poles of the identified damped plants (color crosses)}
 \end{subfigure}
-\caption{\label{fig:test_apa_iff}Experimental results of applying Integral Force Feedback to the APA300ML. Obtained damped plant (\subref{fig:test_apa_identified_damped_plants}) and Root Locus (\subref{fig:test_apa_iff_root_locus})}
+\caption{\label{fig:test_apa_iff}Experimental results of applying Integral Force Feedback to the APA300ML. Obtained damped plant (\subref{fig:test_apa_identified_damped_plants}) and Root Locus (\subref{fig:test_apa_iff_root_locus}) corresponding to the implemented IFF controller \eqref{eq:test_apa_Kiff_formula}}
 \end{figure}
+
+
 \chapter{APA300ML - 2 Degrees of Freedom Model}
-\label{sec:org4cdbff1}
 \label{sec:test_apa_model_2dof}
+In this section, a Simscape model (Figure \ref{fig:test_apa_bench_model}) of the measurement bench is used to tune the 2 degrees of freedom model of the APA using the measured frequency response functions.
 
-In this section, a simscape model (Figure \ref{fig:test_apa_bench_model}) of the measurement bench is used to compare the model of the APA with the measured frequency response functions.
+This 2 degrees of freedom model is developed to accurately represent the APA300ML dynamics while having low complexity and low number of associated states.
-
+After the model presented, the procedure to tune the model is described and the obtained model dynamics is compared with the measurements.
-A 2 degrees of freedom model is used to model the APA300ML.
-This model is presented in Section \ref{ssec:test_apa_2dof_model} and the procedure to tuned the model is described in Section \ref{ssec:test_apa_2dof_model_tuning}.
-The obtained model dynamics is compared with the measurements in Section \ref{ssec:test_apa_2dof_model_result}.
 
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1,width=0.8\linewidth]{figs/test_apa_bench_model.png}
 \caption{\label{fig:test_apa_bench_model}Screenshot of the Simscape model}
 \end{figure}
-\section{Two Degrees of Freedom APA Model}
+
-\label{sec:org5d75a3a}
+\paragraph{Two Degrees of Freedom APA Model}
-\label{ssec:test_apa_2dof_model}
 
 The model of the amplified piezoelectric actuator is shown in Figure \ref{fig:test_apa_2dof_model}.
 It can be decomposed into three components:
@@ -530,9 +534,9 @@ It can be decomposed into three components:
 \item the shell whose axial properties are represented by \(k_1\) and \(c_1\)
 \item the actuator stacks whose contribution in the axial stiffness is represented by \(k_a\) and \(c_a\).
 A force source \(\tau\) represents the axial force induced by the force sensor stacks.
-The gain \(g_a\) (in \(N/m\)) is used to convert the applied voltage \(V_a\) to the axial force \(\tau\)
+The sensitivity \(g_a\) (in \(N/m\)) is used to convert the applied voltage \(V_a\) to the axial force \(\tau\)
 \item the sensor stack whose contribution in the axial stiffness is represented by \(k_e\) and \(c_e\).
-A  sensor measures the stack strain \(d_L\) which is then converted to a voltage \(V_s\) using a gain \(g_s\) (in \(V/m\))
+A sensor measures the stack strain \(d_e\) which is then converted to a voltage \(V_s\) using a sensitivity \(g_s\) (in \(V/m\))
 \end{itemize}
 
 Such simple model has some limitations:
@@ -547,16 +551,13 @@ Such simple model has some limitations:
 \includegraphics[scale=1]{figs/test_apa_2dof_model.png}
 \caption{\label{fig:test_apa_2dof_model}Schematic of the two degrees of freedom model of the APA300ML, adapted from \cite{souleille18_concep_activ_mount_space_applic}}
 \end{figure}
-\section{Tuning of the APA model}
-\label{sec:org10fafb5}
-\label{ssec:test_apa_2dof_model_tuning}
 
-9 parameters (\(m\), \(k_1\), \(c_1\), \(k_e\), \(c_e\), \(k_a\), \(c_a\), \(g_s\) and \(g_a\)) have to be tuned such that the dynamics of the model (Figure \ref{fig:test_apa_2dof_model_simscape}) well represents the identified dynamics in Section \ref{sec:test_apa_dynamics}.
+9 parameters (\(m\), \(k_1\), \(c_1\), \(k_e\), \(c_e\), \(k_a\), \(c_a\), \(g_s\) and \(g_a\)) have to be tuned such that the dynamics of the model (Figure \ref{fig:test_apa_2dof_model_Simscape}) well represents the identified dynamics in Section \ref{sec:test_apa_dynamics}.
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1]{figs/test_apa_2dof_model_simscape.png}
+\includegraphics[scale=1]{figs/test_apa_2dof_model_Simscape.png}
-\caption{\label{fig:test_apa_2dof_model_simscape}Schematic of the two degrees of freedom model of the APA300ML with input \(V_a\) and outputs \(d_e\) and \(V_s\)}
+\caption{\label{fig:test_apa_2dof_model_Simscape}Schematic of the two degrees of freedom model of the APA300ML with input \(V_a\) and outputs \(d_e\) and \(V_s\)}
 \end{figure}
 
 First, the mass \(m\) supported by the APA300ML can be estimated from the geometry and density of the different parts or by directly measuring it using a precise weighing scale.
@@ -582,9 +583,9 @@ Knowing from \eqref{eq:test_apa_tot_stiffness} that the total stiffness is \(k_{
 Then, \(c_a\) (and therefore \(c_e = 2 c_a\)) can be tuned to match the damping ratio of the identified resonance.
 \(c_a = 100\,Ns/m\) and \(c_e = 200\,Ns/m\) are obtained.
 
-Finally, the two gains \(g_s\) and \(g_a\) can be tuned to match the gain of the identified transfer functions.
+Finally \(g_s\) and \(g_a\) can be tuned to match the gain of the identified transfer functions.
 
-The obtained parameters of the model shown in Figure \ref{fig:test_apa_2dof_model_simscape} are summarized in Table \ref{tab:test_apa_2dof_parameters}.
+The obtained parameters of the model shown in Figure \ref{fig:test_apa_2dof_model_Simscape} are summarized in Table \ref{tab:test_apa_2dof_parameters}.
 
 \begin{table}[htbp]
 \centering
@@ -606,9 +607,6 @@ The obtained parameters of the model shown in Figure \ref{fig:test_apa_2dof_mode
 \caption{\label{tab:test_apa_2dof_parameters}Summary of the obtained parameters for the 2 DoF APA300ML model}
 
 \end{table}
-\section{Obtained Dynamics}
-\label{sec:org0831884}
-\label{ssec:test_apa_2dof_model_result}
 
 The dynamics of the two degrees of freedom model of the APA300ML is now extracted using optimized parameters (listed in Table \ref{tab:test_apa_2dof_parameters}) from the Simscape model.
 It is compared with the experimental data in Figure \ref{fig:test_apa_2dof_comp_frf}.
@@ -630,13 +628,12 @@ This indicates that this model represents well the axial dynamics of the APA300M
 \end{subfigure}
 \caption{\label{fig:test_apa_2dof_comp_frf}Comparison of the measured frequency response functions and the identified dynamics from the 2DoF model of the APA300ML. Both for the dynamics from \(u\) to \(d_e\) (\subref{fig:test_apa_2dof_comp_frf_enc}) (\subref{fig:test_apa_2dof_comp_frf_force}) and from \(u\) to \(V_s\) (\subref{fig:test_apa_2dof_comp_frf_force})}
 \end{figure}
-\chapter{APA300ML - Super Element}
-\label{sec:org8292e07}
-\label{sec:test_apa_model_flexible}
 
+\chapter{APA300ML - Super Element}
+\label{sec:test_apa_model_flexible}
 In this section, a \emph{super element} of the APA300ML is computed using a finite element software\footnote{Ansys\textsuperscript{\textregistered} was used}.
 It is then imported in Simscape (in the form of a stiffness matrix and a mass matrix) and included in the same model that was used in \ref{sec:test_apa_model_2dof}.
-This procedure is illustrated in Figure \ref{fig:test_apa_super_element_simscape}.
+This procedure is illustrated in Figure \ref{fig:test_apa_super_element_Simscape}.
 Several \emph{remote points} are defined in the finite element model (here illustrated by colorful planes and numbers from \texttt{1} to \texttt{5}) and are then make accessible in the Simscape model as shown at the right by the ``frames'' \texttt{F1} to \texttt{F5}.
 
 For the APA300ML \emph{super element}, 5 \emph{remote points} are defined.
@@ -647,18 +644,17 @@ It will be used to measure the strain experience by this stack, and model the se
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1,width=1.0\linewidth]{figs/test_apa_super_element_simscape.png}
+\includegraphics[scale=1,width=1.0\linewidth]{figs/test_apa_super_element_Simscape.png}
-\caption{\label{fig:test_apa_super_element_simscape}Finite Element Model of the APA300ML with ``remotes points'' on the left. Simscape model with included ``Reduced Order Flexible Solid'' on the right.}
+\caption{\label{fig:test_apa_super_element_Simscape}Finite Element Model of the APA300ML with ``remotes points'' on the left. Simscape model with included ``Reduced Order Flexible Solid'' on the right.}
 \end{figure}
-\section{Identification of the Actuator and Sensor constants}
+
-\label{sec:org28294e9}
+\paragraph{Identification of the Actuator and Sensor constants}
-\label{ssec:test_apa_flexible_ga_gs}
 
 Once the APA300ML \emph{super element} is included in the Simscape model, the transfer function from \(F_a\) to \(d_L\) and \(d_e\) can be extracted.
 The gains \(g_a\) and \(g_s\) are then be tuned such that the gain of the transfer functions are matching the identified ones.
 By doing so, \(g_s = 4.9\,V/\mu m\) and \(g_a = 23.2\,N/V\) are obtained.
 
-To make sure these ``gains'' are physically valid, it is possible to estimate them from physical properties of the piezoelectric stack material.
+To make sure the sensitivities \(g_a\) and \(g_s\) are physically valid, it is possible to estimate them from physical properties of the piezoelectric stack material.
 
 From \cite[p. 123]{fleming14_desig_model_contr_nanop_system}, the relation between relative displacement \(d_L\) of the sensor stack and generated voltage \(V_s\) is given by \eqref{eq:test_apa_piezo_strain_to_voltage} and from \cite{fleming10_integ_strain_force_feedb_high} the relation between the force \(F_a\) and the applied voltage \(V_a\) is given by \eqref{eq:test_apa_piezo_voltage_to_force}.
 
@@ -690,16 +686,15 @@ From these parameters, \(g_s = 5.1\,V/\mu m\) and \(g_a = 26\,N/V\) were obtaine
 \(n\) & \(160\) per stack & Number of layers in the piezoelectric stack\\
 \bottomrule
 \end{tabularx}
-\caption{\label{tab:test_apa_piezo_properties}Piezoelectric properties used for the estimation of the sensor and actuators ``gains''}
+\caption{\label{tab:test_apa_piezo_properties}Piezoelectric properties used for the estimation of the sensor and actuators sensitivities}
 
 \end{table}
-\section{Comparison of the obtained dynamics}
-\label{sec:org14d2335}
-\label{ssec:test_apa_flexible_comp_frf}
 
-The obtained dynamics using the \emph{super element} with the tuned ``sensor gain'' and ``actuator gain'' are compared with the experimentally identified frequency response functions in Figure \ref{fig:test_apa_super_element_comp_frf}.
+\paragraph{Comparison of the obtained dynamics}
+
+The obtained dynamics using the \emph{super element} with the tuned ``sensor sensitivity'' and ``actuator sensitivity'' are compared with the experimentally identified frequency response functions in Figure \ref{fig:test_apa_super_element_comp_frf}.
 A good match between the model and the experimental results is observed.
-It is however a bit surprising that the model is a bit ``softer'' than the measured system as finite element models are usually overestimating the stiffness.
+It is however surprising that the model is ``softer'' than the measured system as finite element models are usually overestimating the stiffness (see Section \ref{ssec:test_apa_spurious_resonances} for possible explanations).
 
 Using this simple test bench, it can be concluded that the \emph{super element} model of the APA300ML well captures the axial dynamics of the actuator (the actuator stacks, the force sensor stack as well as the shell used as a mechanical lever).
 
@@ -716,13 +711,13 @@ Using this simple test bench, it can be concluded that the \emph{super element}
 \end{center}
 \subcaption{\label{fig:test_apa_super_element_comp_frf_force}from $u$ to $V_s$}
 \end{subfigure}
-\caption{\label{fig:test_apa_super_element_comp_frf}Comparison of the measured frequency response functions and the identified dynamics from the ``flexible'' model of the APA300ML. Both for the dynamics from \(u\) to \(d_e\) (\subref{fig:test_apa_2dof_comp_frf_enc}) (\subref{fig:test_apa_2dof_comp_frf_force}) and from \(u\) to \(V_s\) (\subref{fig:test_apa_2dof_comp_frf_force})}
+\caption{\label{fig:test_apa_super_element_comp_frf}Comparison of the measured frequency response functions and the identified dynamics from the finite element model of the APA300ML. Both for the dynamics from \(u\) to \(d_e\) (\subref{fig:test_apa_2dof_comp_frf_enc}) (\subref{fig:test_apa_2dof_comp_frf_force}) and from \(u\) to \(V_s\) (\subref{fig:test_apa_2dof_comp_frf_force})}
 \end{figure}
+
 \chapter{Conclusion}
-\label{sec:org10e068b}
 \label{sec:test_apa_conclusion}
 
-In this study, the received amplified piezoelectric actuators ``APA300ML'' have been characterized to make sure they are fulfilling all the requirements determined during the detailed design phase.
+In this study, the amplified piezoelectric actuators ``APA300ML'' have been characterized to make sure they are fulfilling all the requirements determined during the detailed design phase.
 
 Geometrical features such as the flatness of its interfaces, electrical capacitance and achievable strokes were measured in Section \ref{sec:test_apa_basic_meas}.
 These simple measurements allowed for early detection of a manufacturing defect in one of the APA300ML.
@@ -733,7 +728,7 @@ Although a non-minimum zero was identified in the transfer function from \(u\) t
 
 Then, two different models were used to represent the APA300ML dynamics.
 In Section \ref{sec:test_apa_model_2dof}, a simple two degrees of freedom mass-spring-damper model was presented and tuned based on the measured dynamics.
-After following a tuning procedure (see Section \ref{ssec:test_apa_2dof_model_tuning}), the model dynamics was shown to match very well with the experiment.
+After following a tuning procedure, the model dynamics was shown to match very well with the experiment.
 However, it is important to note that this model only represents the axial dynamics of the actuators, assuming infinite stiffness in other directions.
 
 In Section \ref{sec:test_apa_model_flexible}, a \emph{super element} extracted from a finite element model was used to model the APA300ML.
@@ -741,5 +736,6 @@ This time, the \emph{super element} represents the dynamics of the APA300ML in a
 However, only the axial dynamics could be compared with the experimental results yielding a good match.
 The benefit of employing this model over the two degrees of freedom model is not immediately apparent due to its increased complexity and the larger number of model states involved.
 Nonetheless, the \emph{super element} model's value will become clear in subsequent sections, when its capacity to accurately model the APA300ML's flexibility across various directions will be important.
+
 \printbibliography[heading=bibintoc,title={Bibliography}]
 \end{document}