Add complementary filters experimental results

phd-thesis.tex

@@ -1,4 +1,4 @@
-% Created 2025-04-20 Sun 18:06
+% Created 2025-04-20 Sun 22:28
 % Intended LaTeX compiler: pdflatex
 \documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
 
@@ -12264,6 +12264,7 @@ Several scientific experiments were replicated, such as:
 \end{itemize}
 
 Unless explicitly stated, all closed-loop experiments were performed using the robust (i.e. conservative) high authority controller designed in Section~\ref{ssec:test_id31_iff_hac_controller}.
+Higher performance controllers using complementary filters are investigated in Section~\ref{ssec:test_id31_cf_control}.
 
 For each experiment, the obtained performances are compared to the specifications for the most demanding case in which nano-focusing optics are used to focus the beam down to \(200\,nm\times 100\,nm\).
 In this case, the goal is to keep the sample's point of interest in the beam, and therefore the \(D_y\) and \(D_z\) positioning errors should be less than \(200\,nm\) and \(100\,nm\) peak-to-peak, respectively.
@@ -12348,12 +12349,11 @@ Nevertheless, even with this robust (i.e. conservative) HAC implementation, the
 
 A comparative analysis was conducted using three tomography scans at \(180\,\text{deg/s}\) to evaluate the effectiveness of the HAC-LAC strategy in reducing positioning errors.
 The scans were performed under three conditions: open-loop, with decentralized IFF control, and with the complete HAC-LAC strategy.
-For these specific measurements, an enhanced high authority controller was optimized for low payload masses to meet the performance requirements.
+For this specific measurement, an enhanced high authority controller (discussed in Section~\ref{ssec:test_id31_cf_control}) was optimized for low payload masses to meet the performance requirements.
 
 Figure~\ref{fig:test_id31_hac_cas_cl} presents the cumulative amplitude spectra of the position errors for all three cases.
 The results reveal two distinct control contributions: the decentralized IFF effectively attenuates vibrations near the nano-hexapod suspension modes (an achievement not possible with HAC alone), while the high authority controller suppresses low-frequency vibrations primarily arising from Spindle guiding errors.
 Notably, the spectral patterns in Figure~\ref{fig:test_id31_hac_cas_cl} closely resemble the cumulative amplitude spectra computed in the project's early stages.
-
 This experiment also illustrates that when needed, performance can be enhanced by designing controllers for specific experimental conditions rather than relying solely on robust controllers that can accommodate all payload ranges.
 
 \begin{figure}[htbp]
@@ -12607,6 +12607,73 @@ Alternatively, a feedforward controller could improve the lateral positioning ac
 \end{subfigure}
 \caption{\label{fig:test_id31_diffraction_tomo}Diffraction tomography scans (combined \(R_z\) and \(D_y\) motions) at several \(D_y\) velocities (\(R_z\) rotational velocity is \(6\,\text{deg/s}\)).}
 \end{figure}
+\subsubsection{Feedback control using Complementary Filters}
+\label{ssec:test_id31_cf_control}
+
+A control architecture utilizing complementary filters to shape the closed-loop transfer functions was proposed during the detail design phase.
+Experimental validation of this architecture using the NASS is presented herein.
+
+Given that performance requirements are specified in the Cartesian frame, decoupling of the plant within this frame was achieved using Jacobian matrices.
+Consequently, the control space comprises the directions \(D_x\), \(D_y\), \(D_z\), \(R_x\), and \(R_y\).
+Control performance in each of these directions can be tuned independently.
+A schematic of the proposed control architecture is illustrated in Figure~\ref{fig:test_id31_cf_control}.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/test_id31_cf_control.png}
+\caption{\label{fig:test_id31_cf_control}Control architecture in the Cartesian frame. Only the controller corresponding to the \(D_z\) direction is shown. \(H_L\) and \(H_H\) are complementary filters.}
+\end{figure}
+
+Implementation of this control architecture necessitates a plant model, which must subsequently be inverted.
+This plant model was derived from the multi-body model incorporating the previously detailed 2-DoF APA model, such that the model order stays relatively low.
+Proposed analytical formulas for complementary filters having \(40\,\text{dB/dec}\) were used during this experimental validation.
+An initial experimental validation was conducted under no-payload conditions, with control applied solely to the \(D_y\), \(D_z\), and \(R_y\) directions.
+Increased control bandwidth was achieved for the \(D_z\) and \(R_y\) directions through appropriate tuning of the parameter \(\omega_0\).
+The experimentally measured closed-loop sensitivity transfer functions corresponding to these three controlled directions are presented in Figure~\ref{fig:test_id31_cf_control_dy_dz_diff}.
+
+Another test was conducted with a \(26\,\text{kg}\) payload.
+For this configuration, complementary filters were implemented with \(\omega_0 = 2\pi \cdot 10\,\text{rad/s}\), and parameter \(\alpha\) was varied.
+The resulting experimentally obtained closed-loop transfer functions are compared against the theoretical complementary filter responses in Figure~\ref{fig:test_id31_cf_control_alpha}.
+As illustrated in the figure, a close correspondence between the measured closed-loop responses and the target complementary filter magnitude was observed.
+It also shows that the parameter \(\alpha\) provides a mechanism for managing the trade-off between low-frequency disturbance rejection performance and the potential amplification of disturbances within the crossover frequency region.
+
+\begin{figure}[htbp]
+\begin{subfigure}{0.49\textwidth}
+\begin{center}
+\includegraphics[scale=1,scale=0.9]{figs/test_id31_cf_control_dy_dz_diff.png}
+\end{center}
+\subcaption{\label{fig:test_id31_cf_control_dy_dz_diff}Chose of bandwidth using $\omega_0$, $m = 0\,\text{kg}$}
+\end{subfigure}
+\begin{subfigure}{0.49\textwidth}
+\begin{center}
+\includegraphics[scale=1,scale=0.9]{figs/test_id31_cf_control_alpha.png}
+\end{center}
+\subcaption{\label{fig:test_id31_cf_control_alpha}Effect of a change of $\alpha$, $m = 26\,\text{kg}$}
+\end{subfigure}
+\caption{\label{fig:test_id31_cf_control_results}Measured closed-loop transfer functions. Different bandwidth can be specified for different directions using \(\omega_0\) (\subref{fig:test_id31_cf_control_dy_dz_diff}). The shape can be adjusted using parameter \(\alpha\) (\subref{fig:test_id31_cf_control_alpha}).}
+\end{figure}
+
+Finally, \(\omega_0\) was gradually increased to estimate the maximum bandwidth (i.e. the best low frequency disturbance rejection) that can be achieved with this architecture.
+No payload was used for this test, and the parameter \(\omega_0\) was increased for the controllers in the \(D_y\) and \(D_z\) directions.
+A value \(\omega_0 = 2\pi \cdot 60 \,\text{rad/s}\) could be achieved.
+Measured closed-loop transfer functions are shown in Figure~\ref{fig:test_id31_high_bandwidth}, indicating a reduction of disturbances in the considered direction of \(1000\) at \(1\,\text{Hz}\).
+For higher values of \(\omega_0\), the system became unstable in the vertical direction, probably because of the resonance at \(250\,\text{Hz}\) that is not well captured with the multi-body model (Figure~\ref{fig:test_id31_hac_plant_effect_mass}).
+
+\begin{figure}[htbp]
+\begin{subfigure}{0.49\textwidth}
+\begin{center}
+\includegraphics[scale=1,scale=0.9]{figs/test_id31_high_bandwidth_S.png}
+\end{center}
+\subcaption{\label{fig:test_id31_high_bandwidth_S}Sensitivity}
+\end{subfigure}
+\begin{subfigure}{0.49\textwidth}
+\begin{center}
+\includegraphics[scale=1,scale=0.9]{figs/test_id31_high_bandwidth_T.png}
+\end{center}
+\subcaption{\label{fig:test_id31_high_bandwidth_T}Complementary Sensitivity}
+\end{subfigure}
+\caption{\label{fig:test_id31_high_bandwidth}Measured Closed-Loop Sensitivity (\subref{fig:test_id31_high_bandwidth_S}) and Complementary Sensitivity (\subref{fig:test_id31_high_bandwidth_T}) transfer functions for the highest test bandwidth \(\omega_0 = 2\pi\cdot 60\,\text{rad/s}\).}
+\end{figure}
 \subsubsection*{Conclusion}
 \label{ssec:test_id31_scans_conclusion}