Add closed-loop shaping section

2025-04-03 16:29:57 +02:00 · 2025-04-03 16:29:57 +02:00 · e080db4bec
commit e080db4bec
parent f3bf64da95
41 changed files with 4202 additions and 34 deletions
--- a/figs/detail_control_bode_Kfb.pdf
+++ b/figs/detail_control_bode_Kfb.pdf
--- a/figs/detail_control_bode_Kfb.png
+++ b/figs/detail_control_bode_Kfb.png
--- a/figs/detail_control_bode_plot_loop_gain_robustness.pdf
+++ b/figs/detail_control_bode_plot_loop_gain_robustness.pdf
--- a/figs/detail_control_bode_plot_loop_gain_robustness.png
+++ b/figs/detail_control_bode_plot_loop_gain_robustness.png
--- a/figs/detail_control_bode_plot_mech_sys.pdf
+++ b/figs/detail_control_bode_plot_mech_sys.pdf
--- a/figs/detail_control_bode_plot_mech_sys.png
+++ b/figs/detail_control_bode_plot_mech_sys.png
--- a/figs/detail_control_hinf_filters_result_weights.pdf
+++ b/figs/detail_control_hinf_filters_result_weights.pdf
--- a/figs/detail_control_hinf_filters_result_weights.png
+++ b/figs/detail_control_hinf_filters_result_weights.png
--- a/figs/detail_control_input_uncertainty.pdf
+++ b/figs/detail_control_input_uncertainty.pdf
--- a/figs/detail_control_input_uncertainty.png
+++ b/figs/detail_control_input_uncertainty.png
--- a/figs/detail_control_input_uncertainty.svg
+++ b/figs/detail_control_input_uncertainty.svg
@ -0,0 +1,97 @@
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--- a/figs/detail_control_mech_sys_alone.pdf
+++ b/figs/detail_control_mech_sys_alone.pdf
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--- a/figs/detail_control_nyquist_robustness.pdf
+++ b/figs/detail_control_nyquist_robustness.pdf
--- a/figs/detail_control_nyquist_robustness.png
+++ b/figs/detail_control_nyquist_robustness.png
--- a/figs/detail_control_robust_perf.pdf
+++ b/figs/detail_control_robust_perf.pdf
--- a/figs/detail_control_robust_perf.png
+++ b/figs/detail_control_robust_perf.png
--- a/figs/detail_control_sf_arch.pdf
+++ b/figs/detail_control_sf_arch.pdf
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+++ b/figs/detail_control_sf_arch.png
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--- a/figs/detail_control_sf_arch_class_prefilter.pdf
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--- a/figs/detail_control_sf_arch_eq.pdf
+++ b/figs/detail_control_sf_arch_eq.pdf
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--- a/figs/detail_control_spec_S_T.pdf
+++ b/figs/detail_control_spec_S_T.pdf
--- a/figs/detail_control_spec_S_T.png
+++ b/figs/detail_control_spec_S_T.png
--- a/matlab/src/plotMagUncertainty.m
+++ b/matlab/src/plotMagUncertainty.m
@ -0,0 +1,42 @@
  function [p] = plotMagUncertainty(W, freqs, args)
  % plotMagUncertainty -
  %
  % Syntax: [p] = plotMagUncertainty(W, freqs, args)
  %
  % Inputs:
  %    - W     - Multiplicative Uncertainty Weight
  %    - freqs - Frequency Vector [Hz]
  %    - args  - Optional Arguments:
  %      - G
  %      - color_i
  %      - opacity
  %
  % Outputs:
  %    - p - Plot Handle
      arguments
      W
      freqs double {mustBeNumeric, mustBeNonnegative}
      args.G = tf(1)
      args.color_i (1,1) double {mustBeInteger, mustBeNonnegative} = 0
      args.opacity (1,1) double {mustBeNumeric, mustBeNonnegative} = 0.3
      args.DisplayName char = ''
  end
  % Get defaults colors
  colors = get(groot, 'defaultAxesColorOrder');
  p = patch([freqs flip(freqs)], ...
            [abs(squeeze(freqresp(args.G, freqs, 'Hz'))).*(1 + abs(squeeze(freqresp(W, freqs, 'Hz')))); ...
             flip(abs(squeeze(freqresp(args.G, freqs, 'Hz'))).*max(1 - abs(squeeze(freqresp(W, freqs, 'Hz'))), 1e-6))], 'w', ...
            'DisplayName', args.DisplayName);
  if args.color_i == 0
      p.FaceColor = [0; 0; 0];
  else
      p.FaceColor = colors(args.color_i, :);
  end
  p.EdgeColor = 'none';
  p.FaceAlpha = args.opacity;
   end
--- a/matlab/src/plotPhaseUncertainty.m
+++ b/matlab/src/plotPhaseUncertainty.m
@ -0,0 +1,47 @@
  function [p] = plotPhaseUncertainty(W, freqs, args)
  % plotPhaseUncertainty -
  %
  % Syntax: [p] = plotPhaseUncertainty(W, freqs, args)
  %
  % Inputs:
  %    - W     - Multiplicative Uncertainty Weight
  %    - freqs - Frequency Vector [Hz]
  %    - args  - Optional Arguments:
  %      - G
  %      - color_i
  %      - opacity
  %
  % Outputs:
  %    - p - Plot Handle
  arguments
      W
      freqs double {mustBeNumeric, mustBeNonnegative}
      args.G = tf(1)
      args.color_i (1,1) double {mustBeInteger, mustBeNonnegative} = 0
      args.opacity (1,1) double {mustBeNumeric, mustBePositive} = 0.3
      args.DisplayName char = ''
  end
  % Get defaults colors
  colors = get(groot, 'defaultAxesColorOrder');
  % Compute Phase Uncertainty
  Dphi = 180/pi*asin(abs(squeeze(freqresp(W, freqs, 'Hz'))));
  Dphi(abs(squeeze(freqresp(W, freqs, 'Hz'))) > 1) = 360;
  % Compute Plant Phase
  G_ang = 180/pi*angle(squeeze(freqresp(args.G, freqs, 'Hz')));
  p = patch([freqs flip(freqs)], [G_ang+Dphi; flip(G_ang-Dphi)], 'w', ...
            'DisplayName', args.DisplayName);
  if args.color_i == 0
      p.FaceColor = [0; 0; 0];
  else
      p.FaceColor = colors(args.color_i, :);
  end
  p.EdgeColor = 'none';
  p.FaceAlpha = args.opacity;
  end
--- a/nass-control.bib
+++ b/nass-control.bib
@ -424,3 +424,41 @@
  keywords        = {complementary filters},
 }
@article{oomen18_advan_motion_contr_precis_mechat,
  author          = {Tom Oomen},
  title           = {Advanced Motion Control for Precision Mechatronics:
                  Control, Identification, and Learning of Complex Systems},
  journal         = {IEEJ Journal of Industry Applications},
  volume          = 7,
  number          = 2,
  pages           = {127-140},
  year            = 2018,
  doi             = {10.1541/ieejjia.7.127},
  url             = {https://doi.org/10.1541/ieejjia.7.127},
  keywords        = {favorite},
 }
@inproceedings{collette14_vibrat,
  author          = {Collette, C. and Matichard, F},
  title           = {Vibration control of flexible structures using fusion of
                  inertial sensors and hyper-stable actuator-sensor pairs},
  booktitle       = {International Conference on Noise and Vibration Engineering
                  (ISMA2014)},
  year            = 2014,
  keywords        = {sensor fusion},
 }
@techreport{bibel92_guidel_h,
  author          = {Bibel, John E and Malyevac, D Stephen},
  institution     = {NAVAL SURFACE WARFARE CENTER DAHLGREN DIV VA},
  title           = {Guidelines for the selection of weighting functions for
                  H-infinity control},
  year            = 1992,
 }
--- a/nass-control.org
+++ b/nass-control.org
--- a/nass-control.pdf
+++ b/nass-control.pdf
--- a/nass-control.tex
+++ b/nass-control.tex
@ -1,4 +1,4 @@
-% Created 2025-04-03 Thu 15:24
+% Created 2025-04-03 Thu 17:13
 % Intended LaTeX compiler: pdflatex
 \documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
@ -25,6 +25,7 @@ Several considerations:
 \item Section \ref{sec:detail_control_optimization}: How to design the controller
 \end{itemize}
 \chapter{Multiple Sensor Control}
 \label{sec:orgacbb166}
 \label{sec:detail_control_multiple_sensor}
 \textbf{Look at what was done in the introduction \href{file:///home/thomas/Cloud/work-projects/ID31-NASS/phd-thesis-chapters/A0-nass-introduction/nass-introduction.org}{Stewart platforms: Control architecture}}
@ -142,12 +143,13 @@ Although many design methods of complementary filters have been proposed in the
 Fortunately, both the robustness of the fusion and the super sensor characteristics can be linked to the magnitude of the complementary filters \cite{dehaeze19_compl_filter_shapin_using_synth}.
 Based on that, this paper introduces a new way to design complementary filters using the \(\mathcal{H}_\infty\) synthesis which allows to shape the complementary filters' magnitude in an easy and intuitive way.
 \section{Sensor Fusion and Complementary Filters Requirements}
-\label{sec:detail_control_sensor_fusion_requirements}
+\label{sec:org9733630}
 \label{ssec:detail_control_sensor_fusion_requirements}
 Complementary filtering provides a framework for fusing signals from different sensors.
 As the effectiveness of the fusion depends on the proper design of the complementary filters, they are expected to fulfill certain requirements.
 These requirements are discussed in this section.
 \subsection{Sensor Fusion Architecture}
-\label{sec:detail_control_sensor_fusion}
+\label{sec:org45e5dc5}
 A general sensor fusion architecture using complementary filters is shown in Fig. \ref{fig:detail_control_sensor_fusion_overview} where several sensors (here two) are measuring the same physical quantity \(x\).
 The two sensors output signals \(\hat{x}_1\) and \(\hat{x}_2\) are estimates of \(x\).
@ -172,7 +174,7 @@ Therefore, a pair of complementary filter needs to satisfy the following conditi
 It will soon become clear why the complementary property is important for the sensor fusion architecture.
 \subsection{Sensor Models and Sensor Normalization}
-\label{sec:detail_control_sensor_models}
+\label{sec:org117d609}
 In order to study such sensor fusion architecture, a model for the sensors is required.
 Such model is shown in Fig. \ref{fig:detail_control_sensor_model} and consists of a linear time invariant (LTI) system \(G_i(s)\) representing the sensor dynamics and an input \(n_i\) representing the sensor noise.
@ -216,7 +218,7 @@ The super sensor output is therefore equal to:
 \caption{\label{fig:detail_control_fusion_super_sensor}Sensor fusion architecture with two normalized sensors.}
 \end{figure}
 \subsection{Noise Sensor Filtering}
-\label{sec:detail_control_noise_filtering}
+\label{sec:orgb1cc291}
 In this section, it is supposed that all the sensors are perfectly normalized, such that:
@ -255,7 +257,7 @@ However, the two sensors have usually high noise levels over distinct frequency
 In such case, to lower the noise of the super sensor, the norm \(|H_1(j\omega)|\) has to be small when \(\Phi_{n_1}(\omega)\) is larger than \(\Phi_{n_2}(\omega)\) and the norm \(|H_2(j\omega)|\) has to be small when \(\Phi_{n_2}(\omega)\) is larger than \(\Phi_{n_1}(\omega)\).
 Hence, by properly shaping the norm of the complementary filters, it is possible to reduce the noise of the super sensor.
 \subsection{Sensor Fusion Robustness}
-\label{sec:detail_control_fusion_robustness}
+\label{sec:orgc1bccc3}
 In practical systems the sensor normalization is not perfect and condition \eqref{eq:detail_control_perfect_dynamics} is not verified.
@ -315,14 +317,15 @@ For instance, the phase \(\Delta\phi(\omega)\) added by the super sensor dynamic
 As it is generally desired to limit the maximum phase added by the super sensor, \(H_1(s)\) and \(H_2(s)\) should be designed such that \(\Delta \phi\) is bounded to acceptable values.
 Typically, the norm of the complementary filter \(|H_i(j\omega)|\) should be made small when \(|w_i(j\omega)|\) is large, i.e., at frequencies where the sensor dynamics is uncertain.
 \section{Complementary Filters Shaping}
-\label{sec:detail_control_hinf_method}
+\label{sec:org48106a8}
-As shown in Section \ref{sec:detail_control_sensor_fusion_requirements}, the noise and robustness of the super sensor are a function of the complementary filters' norm.
+\label{ssec:detail_control_hinf_method}
 As shown in Section \ref{ssec:detail_control_sensor_fusion_requirements}, the noise and robustness of the super sensor are a function of the complementary filters' norm.
 Therefore, a synthesis method of complementary filters that allows to shape their norm would be of great use.
 In this section, such synthesis is proposed by writing the synthesis objective as a standard \(\mathcal{H}_\infty\) optimization problem.
 As weighting functions are used to represent the wanted complementary filters' shape during the synthesis, their proper design is discussed.
 Finally, the synthesis method is validated on an simple example.
 \subsection{Synthesis Objective}
-\label{sec:detail_control_synthesis_objective}
+\label{sec:org8a4a881}
 The synthesis objective is to shape the norm of two filters \(H_1(s)\) and \(H_2(s)\) while ensuring their complementary property \eqref{eq:detail_control_comp_filter}.
 This is equivalent as to finding proper and stable transfer functions \(H_1(s)\) and \(H_2(s)\) such that conditions \eqref{eq:detail_control_hinf_cond_complementarity}, \eqref{eq:detail_control_hinf_cond_h1} and \eqref{eq:detail_control_hinf_cond_h2} are satisfied.
@ -337,7 +340,7 @@ This is equivalent as to finding proper and stable transfer functions \(H_1(s)\)
 \(W_1(s)\) and \(W_2(s)\) are two weighting transfer functions that are carefully chosen to specify the maximum wanted norm of the complementary filters during the synthesis.
 \subsection{Shaping of Complementary Filters using \(\mathcal{H}_\infty\) synthesis}
-\label{sec:detail_control_hinf_synthesis}
+\label{sec:org2122803}
 In this section, it is shown that the synthesis objective can be easily expressed as a standard \(\mathcal{H}_\infty\) optimization problem and therefore solved using convenient tools readily available.
@ -385,7 +388,7 @@ Note that there is only an implication between the \(\mathcal{H}_\infty\) norm c
 Hence, the optimization may be a little bit conservative with respect to the set of filters on which it is performed, see \cite[Chap. 2.8.3]{skogestad07_multiv_feedb_contr}.
 In practice, this is however not an found to be an issue.
 \subsection{Weighting Functions Design}
-\label{sec:detail_control_hinf_weighting_func}
+\label{sec:orgaaf1d4a}
 Weighting functions are used during the synthesis to specify the maximum allowed complementary filters' norm.
 The proper design of these weighting functions is of primary importance for the success of the presented \(\mathcal{H}_\infty\) synthesis of complementary filters.
@ -431,7 +434,7 @@ The typical magnitude of a weighting function generated using \eqref{eq:detail_c
 \caption{\label{fig:detail_control_weight_formula}Magnitude of a weighting function generated using formula \eqref{eq:detail_control_weight_formula}, \(G_0 = 1e^{-3}\), \(G_\infty = 10\), \(\omega_c = \SI{10}{Hz}\), \(G_c = 2\), \(n = 3\).}
 \end{figure}
 \subsection{Validation of the proposed synthesis method}
-\label{sec:detail_control_hinf_example}
+\label{sec:org6a4ff6e}
 The proposed methodology for the design of complementary filters is now applied on a simple example.
 Let's suppose two complementary filters \(H_1(s)\) and \(H_2(s)\) have to be designed such that:
@ -500,7 +503,8 @@ As expected, the obtained filters are of order \(5\), that is the sum of the wei
 This simple example illustrates the fact that the proposed methodology for complementary filters shaping is easy to use and effective.
 A more complex real life example is taken up in the next section.
 \section{``Closed-Loop'' complementary filters}
-\label{sec:detail_control_closed_loop_complementary_filters}
+\label{sec:org8c07343}
 \label{ssec:detail_control_closed_loop_complementary_filters}
 An alternative way to implement complementary filters is by using a fundamental property of the classical feedback architecture shown in Fig. \ref{fig:detail_control_feedback_sensor_fusion}.
 This idea is discussed in \cite{mahony05_compl_filter_desig_special_orthog,plummer06_optim_compl_filter_their_applic_motion_measur,jensen13_basic_uas}.
@ -581,7 +585,7 @@ As an example, two ``closed-loop'' complementary filters are designed using the
 The weighting functions are designed using formula \eqref{eq:detail_control_weight_formula} with parameters shown in Table \ref{tab:detail_control_weights_params}.
 After synthesis, a filter \(L(s)\) is obtained whose magnitude is shown in Fig. \ref{fig:detail_control_hinf_filters_results_mixed_sensitivity} by the black dashed line.
 The ``closed-loop'' complementary filters are compared with the inverse magnitude of the weighting functions in Fig. \ref{fig:detail_control_hinf_filters_results_mixed_sensitivity} confirming that the synthesis is successful.
-The obtained ``closed-loop'' complementary filters are indeed equal to the ones obtained in Section \ref{sec:detail_control_hinf_example}.
+The obtained ``closed-loop'' complementary filters are indeed equal to the ones obtained in Section \ref{ssec:detail_control_hinf_method}.
 \begin{figure}[htbp]
 \centering
@ -589,6 +593,7 @@ The obtained ``closed-loop'' complementary filters are indeed equal to the ones
 \caption{\label{fig:detail_control_hinf_filters_results_mixed_sensitivity}Bode plot of the obtained complementary filters after \(\mathcal{H}_\infty\) mixed-sensitivity synthesis}
 \end{figure}
 \section{Synthesis of a set of three complementary filters}
 \label{sec:org7f588fb}
 \label{sec:detail_control_hinf_three_comp_filters}
 Some applications may require to merge more than two sensors \cite{stoten01_fusion_kinet_data_using_compos_filter,fonseca15_compl}.
@ -628,7 +633,7 @@ The synthesis objective is to compute a set of \(n\) stable transfer functions \
 \([W_1(s),\ W_2(s),\ \dots,\ W_n(s)]\) are weighting transfer functions that are chosen to specify the maximum complementary filters' norm during the synthesis.
-Such synthesis objective is closely related to the one described in Section \ref{sec:detail_control_synthesis_objective}, and indeed the proposed synthesis method is a generalization of the one presented in Section \ref{sec:detail_control_hinf_synthesis}.
+Such synthesis objective is closely related to the one described in Section \ref{ssec:detail_control_hinf_method}, and indeed the proposed synthesis method is a generalization of the one previously presented.
 A set of \(n\) complementary filters can be shaped by applying the standard \(\mathcal{H}_\infty\) synthesis to the generalized plant \(P_n(s)\) described by \eqref{eq:detail_control_generalized_plant_n_filters}.
@ -700,6 +705,7 @@ Figure \ref{fig:detail_control_three_complementary_filters_results} displays the
 \caption{\label{fig:detail_control_three_complementary_filters_results}Bode plot of the inverse weighting functions and of the three complementary filters obtained using the \(\mathcal{H}_\infty\) synthesis}
 \end{figure}
 \section*{Conclusion}
 \label{sec:org9b71274}
 A new method for designing complementary filters using the \(\mathcal{H}_\infty\) synthesis has been proposed.
 It allows to shape the magnitude of the filters by the use of weighting functions during the synthesis.
 This is very valuable in practice as the characteristics of the super sensor are linked to the complementary filters' magnitude.
@ -709,6 +715,7 @@ Several examples were used to emphasize the simplicity and the effectiveness of
 However, the shaping of the complementary filters' magnitude does not allow to directly optimize the super sensor noise and dynamical characteristics.
 Future work will aim at developing a complementary filter synthesis method that minimizes the super sensor noise while ensuring the robustness of the fusion.
 \chapter{Decoupling Strategies}
 \label{sec:orgb61bade}
 \label{sec:detail_control_decoupling}
 When dealing with MIMO systems, a typical strategy is to:
@ -734,26 +741,33 @@ Review of decoupling strategies for Stewart platforms:
 Maybe simpler.
 \end{itemize}
 \section{Interaction Analysis}
 \label{sec:org4e38e50}
 \section{Decentralized Control (actuator frame)}
 \label{sec:orgbcd61c7}
 \section{Center of Stiffness and center of Mass}
 \label{sec:orgb312846}
 \begin{itemize}
 \item Example
 \item Show
 \end{itemize}
 \section{Modal Decoupling}
 \label{sec:orgd3c8cf1}
 \section{Data Based Decoupling}
 \label{sec:orgc72c6af}
 \begin{itemize}
 \item Static decoupling
 \item SVD
 \end{itemize}
 \section*{Conclusion}
 \label{sec:org3802e66}
 Table that compares all the strategies.
 \chapter{Closed-Loop Shaping using Complementary Filters}
 \label{sec:orgdd912f0}
 \label{sec:detail_control_optimization}
 Performance of a feedback control is dictated by closed-loop transfer functions.
@ -780,8 +794,461 @@ Other strategy: Model Based Design:
 In this section, an alternative is proposed in which complementary filters are used for closed-loop shaping.
 It is presented for a SISO system, but can be generalized to MIMO if decoupling is sufficient.
 It will be experimentally demonstrated with the NASS.
 \textbf{Paper's introduction}:
 \textbf{Model based control}
 \textbf{SISO control design methods}
 \begin{itemize}
 \item frequency domain techniques
 \item manual loop-shaping - key idea: modification of the controller such that the open-loop is made according to specifications \cite{oomen18_advan_motion_contr_precis_mechat}.
 \end{itemize}
 This works well because the open loop transfer function is linearly dependent of the controller.
 However, the specifications are given in terms of the final system performance, i.e. as closed-loop specifications.
 \textbf{Norm-based control}
 \(\hinf\) loop-shaping \cite{skogestad07_multiv_feedb_contr}. Far from standard in industry as it requires lot of efforts.
 Problem of robustness to plant uncertainty:
 \begin{itemize}
 \item Trade off performance / robustness. Difficult to obtain high performance in presence of high uncertainty.
 \item Robust control \(\mu\text{-synthesis}\). Takes a lot of effort to model the plant uncertainty.
 \item Sensor fusion: combines two sensors using complementary filters. The high frequency sensor is collocated with the actuator in order to ensure the stability of the system even in presence of uncertainty. \cite{collette15_sensor_fusion_method_high_perfor,collette14_vibrat}
 \end{itemize}
 Complementary filters: \cite{hua05_low_ligo}.
 In this paper, we propose a new controller synthesis method
 \begin{itemize}
 \item based on the use of complementary high pass and low pass filters
 \item inverse based control
 \item direct translation of requirements such as disturbance rejection and robustness to plant uncertainty
 \end{itemize}
 \section{Control Architecture}
 \label{sec:orgbad19e8}
 \label{ssec:detail_control_control_arch}
 \paragraph{Virtual Sensor Fusion}
 \label{sec:org5db8fac}
 Let's consider the control architecture represented in Fig. \ref{fig:detail_control_sf_arch} where \(G^\prime\) is the physical plant to control, \(G\) is a model of the plant, \(k\) is a gain, \(H_L\) and \(H_H\) are complementary filters (\(H_L + H_H = 1\) in the complex sense).
 The signals are the reference signal \(r\), the output perturbation \(d_y\), the measurement noise \(n\) and the control input \(u\).
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1]{figs/detail_control_sf_arch.png}
 \caption{\label{fig:detail_control_sf_arch}Sensor Fusion Architecture}
 \end{figure}
 The dynamics of the closed-loop system is described by the following equations
 \begin{alignat}{5}
 y &= \frac{1+kGH_H}{1+L} dy &&+ \frac{kG^{\prime}}{1+L} r &&- \frac{kG^{\prime}H_L}{1+L} n \\
 u &= -\frac{kH_L}{1+L}   dy &&+ \frac{k}{1+L}           r &&- \frac{kH_L}{1+L}  n
 \end{alignat}
 with \(L = k(G H_H + G^\prime H_L)\).
 The idea of using such architecture comes from sensor fusion \cite{collette14_vibrat,collette15_sensor_fusion_method_high_perfor} where we use two sensors.
 One is measuring the quantity that is required to control, the other is collocated with the actuator in such a way that stability is guaranteed.
 The first one is low pass filtered in order to obtain good performance at low frequencies and the second one is high pass filtered to benefits from its good dynamical properties.
 Here, the second sensor is replaced by a model \(G\) of the plant which is assumed to be stable and minimum phase.
 One may think that the control architecture shown in Fig. \ref{fig:detail_control_sf_arch} is a multi-loop system, but because no non-linear saturation-type element is present in the inner-loop (containing \(k\), \(G\) and \(H_H\) which are all numerically implemented), the structure is equivalent to the architecture shown in Fig. \ref{fig:detail_control_sf_arch_eq}.
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1]{figs/detail_control_sf_arch_eq.png}
 \caption{\label{fig:detail_control_sf_arch_eq}Equivalent feedback architecture}
 \end{figure}
 The dynamics of the system can be rewritten as follow
 \begin{alignat}{5}
 y &=  \frac{1}{1+G^{\prime} K H_L}     dy &&+ \frac{G^{\prime} K}{1+G^{\prime} K H_L} r &&- \frac{G^{\prime} K H_L}{1+G^{\prime} K H_L} n \\
 u &= \frac{-K H_L}{1+G^{\prime} K H_L} dy &&+ \frac{K}{1+G^{\prime} K H_L}            r &&- \frac{K H_L}{1+G^{\prime} K H_L}   n
 \end{alignat}
 with \(K = \frac{k}{1 + H_H G k}\)
 \paragraph{Asymptotic behavior}
 \label{sec:orgb964791}
 We now want to study the asymptotic system obtained when using very high value of \(k\)
 \begin{equation}
  \lim_{k\to\infty} K = \lim_{k\to\infty} \frac{k}{1+H_H G k} = \left( H_H G \right)^{-1}
 \end{equation}
 If the obtained \(K\) is improper, a low pass filter can be added to have its causal realization.
 Also, we want \(K\) to be stable, so \(G\) and \(H_H\) must be minimum phase transfer functions.
 For now on, we will consider the resulting control architecture as shown on Fig. \ref{fig:detail_control_sf_arch_class} where the only ``tuning parameters'' are the complementary filters.
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1]{figs/detail_control_sf_arch_class.png}
 \caption{\label{fig:detail_control_sf_arch_class}Equivalent classical feedback control architecture}
 \end{figure}
 The equations describing the dynamics of the closed-loop system are
 \begin{align}
  y &= \frac{ H_H         dy + G^{\prime} G^{-1} r - G^{\prime} G^{-1} H_L n }{H_H + G^\prime G^{-1} H_L} \label{eq:detail_control_cl_system_y}\\
  u &= \frac{ -G^{-1} H_L dy + G^{-1}            r - G^{-1} H_L            n }{H_H + G^\prime G^{-1} H_L} \label{eq:detail_control_cl_system_u}
 \end{align}
 At frequencies where the model is accurate: \(G^{-1} G^{\prime} \approx 1\), \(H_H + G^\prime G^{-1} H_L \approx H_H + H_L = 1\) and
 \begin{align}
 y        &=  H_H        dy +         r - H_L         n \label{eq:detail_control_cl_performance_y} \\
 u        &= -G^{-1} H_L dy + G^{-1}  r - G^{-1} H_L  n \label{eq:detail_control_cl_performance_u}
 \end{align}
 We obtain a sensitivity transfer function equals to the high pass filter \(S = \frac{y}{dy} = H_H\) and a transmissibility transfer function equals to the low pass filter \(T = \frac{y}{n} = H_L\).
 Assuming that we have a good model of the plant, we have then that the closed-loop behavior of the system converges to the designed complementary filters.
 \section{Translating the performance requirements into the shapes of the complementary filters}
 \label{sec:org1e41339}
 \label{ssec:detail_control_trans_perf}
 The required performance specifications in a feedback system can usually be translated into requirements on the upper bounds of \(\abs{S(j\w)}\) and \(|T(j\omega)|\) \cite{bibel92_guidel_h}.
 The process of designing a controller \(K(s)\) in order to obtain the desired shapes of \(\abs{S(j\w)}\) and \(\abs{T(j\w)}\) is called loop shaping.
 The equations \eqref{eq:detail_control_cl_system_y} and \eqref{eq:detail_control_cl_system_u} describing the dynamics of the studied feedback architecture are not written in terms of \(K\) but in terms of the complementary filters \(H_L\) and \(H_H\).
 In this section, we then translate the typical specifications into the desired shapes of the complementary filters \(H_L\) and \(H_H\).\\
 \paragraph{Nominal Stability (NS)}
 \label{sec:org747119c}
 The closed-loop system is stable if all its elements are stable (\(K\), \(G^\prime\) and \(H_L\)) and if the sensitivity function (\(S = \frac{1}{1 + G^\prime K H_L}\)) is stable.
 For the nominal system (\(G^\prime = G\)), we have \(S = H_H\).
 Nominal stability is then guaranteed if \(H_L\), \(H_H\) and \(G\) are stable and if \(G\) and \(H_H\) are minimum phase (to have \(K\) stable).
 Thus we must design stable and minimum phase complementary filters.\\
 \paragraph{Nominal Performance (NP)}
 \label{sec:org7e3c875}
 Typical performance specifications can usually be translated into upper bounds on \(|S(j\omega)|\) and \(|T(j\omega)|\).
 Two performance weights \(w_H\) and \(w_L\) are defined in such a way that performance specifications are satisfied if
 \begin{equation}
  |w_H(j\omega) S(j\omega)| \le 1,\ |w_L(j\omega) T(j\omega)| \le 1 \quad \forall\omega
 \end{equation}
 For the nominal system, we have \(S = H_H\) and \(T = H_L\), and then nominal performance is ensured by requiring
 \begin{subnumcases}{\text{NP} \Leftrightarrow}\label{eq:detail_control_nominal_performance}
  |w_H(j\omega) H_H(j\omega)| \le 1 \quad \forall\omega \label{eq:detail_control_nominal_perf_hh}\\
  |w_L(j\omega) H_L(j\omega)| \le 1 \quad \forall\omega \label{eq:detail_control_nominal_perf_hl}
 \end{subnumcases}
 The translation of typical performance requirements on the shapes of the complementary filters is discussed below:
 \begin{itemize}
 \item for disturbance rejections, make \(|S| = |H_H|\) small
 \item for noise attenuation, make \(|T| = |H_L|\) small
 \item for control energy reduction, make \(|KS| = |G^{-1}|\) small
 \end{itemize}
 We may have other requirements in terms of stability margins, maximum or minimum closed-loop bandwidth.\\
 \paragraph{Closed-Loop Bandwidth}
 \label{sec:org91bb5c7}
 The closed-loop bandwidth \(\w_B\) can be defined as the frequency where \(\abs{S(j\w)}\) first crosses \(\frac{1}{\sqrt{2}}\) from below.
 If one wants the closed-loop bandwidth to be at least \(\w_B^*\) (e.g. to stabilize an unstable pole), one can required that \(|S(j\omega)| \le \frac{1}{\sqrt{2}}\) below \(\omega_B^*\) by designing \(w_H\) such that \(|w_H(j\omega)| \ge \sqrt{2}\) for \(\omega \le \omega_B^*\).
 Similarly, if one wants the closed-loop bandwidth to be less than \(\w_B^*\), one can approximately require that the magnitude of \(T\) is less than \(\frac{1}{\sqrt{2}}\) at frequencies above \(\w_B^*\) by designing \(w_L\) such that \(|w_L(j\omega)| \ge \sqrt{2}\) for \(\omega \ge \omega_B^*\).\\
 \paragraph{Classical stability margins}
 \label{sec:org17bfb04}
 Gain margin (GM) and phase margin (PM) are usual specifications on controlled system.
 Minimum GM and PM can be guaranteed by limiting the maximum magnitude of the sensibility function \(M_S = \max_{\omega} |S(j\omega)|\):
 \begin{equation}
  \text{GM} \geq \frac{M_S}{M_S-1}; \quad \text{PM} \geq \frac{1}{M_S}
 \end{equation}
 Thus, having \(M_S \le 2\) guarantees a gain margin of at least \(2\) and a phase margin of at least \(\SI{29}{\degree}\).
 For the nominal system \(M_S = \max_\omega |S| = \max_\omega |H_H|\), so one can design \(w_H\) so that \(|w_H(j\omega)| \ge 1/2\) in order to have
 \begin{equation}
  |H_H(j\omega)| \le 2 \quad \forall\omega
 \end{equation}
 and thus obtain acceptable stability margins.\\
 \paragraph{Response time to change of reference signal}
 \label{sec:org555bdc0}
 For the nominal system, the model is accurate and the transfer function from reference signal \(r\) to output \(y\) is \(1\) \eqref{eq:detail_control_cl_performance_y} and does not depends of the complementary filters.
 However, one can add a pre-filter as shown in Fig. \ref{fig:detail_control_sf_arch_class_prefilter}.
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1]{figs/detail_control_sf_arch_class_prefilter.png}
 \caption{\label{fig:detail_control_sf_arch_class_prefilter}Prefilter used to limit input usage}
 \end{figure}
 The transfer function from \(y\) to \(r\) becomes \(\frac{y}{r} = K_r\) and \(K_r\) can we chosen to obtain acceptable response to change of the reference signal.
 Typically, \(K_r\) is a low pass filter of the form
 \begin{equation}
  K_r(s) = \frac{1}{1 + \tau s}
 \end{equation}
 with \(\tau\) corresponding to the desired response time.\\
 \paragraph{Input usage}
 \label{sec:orgaed43be}
 Input usage due to disturbances \(d_y\) and measurement noise \(n\) is determined by \(\big|\frac{u}{d_y}\big| = \big|\frac{u}{n}\big| = \big|G^{-1}H_L\big|\).
 Thus it can be limited by setting an upper bound on \(|H_L|\).
 Input usage due to reference signal \(r\) is determined by \(\big|\frac{u}{r}\big| = \big|G^{-1} K_r\big|\) when using a pre-filter (Fig. \ref{fig:detail_control_sf_arch_class_prefilter}) and \(\big|\frac{u}{r}\big| = \big|G^{-1}\big|\) otherwise.
 Proper choice of \(|K_r|\) is then useful to limit input usage due to change of reference signal.\\
 \paragraph{Robust Stability (RS)}
 \label{sec:org3210a2c}
 Robustness stability represents the ability of the control system to remain stable even though there are differences between the actual system \(G^\prime\) and the model \(G\) that was used to design the controller.
 These differences can have various origins such as unmodelled dynamics or non-linearities.
 To represent the differences between the model and the actual system, one can choose to use the general input multiplicative uncertainty as represented in Fig. \ref{fig:detail_control_input_uncertainty}.
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1]{figs/detail_control_input_uncertainty.png}
 \caption{\label{fig:detail_control_input_uncertainty}Input multiplicative uncertainty}
 \end{figure}
 Then, the set of possible perturbed plant is described by
 \begin{equation}\label{eq:detail_control_multiplicative_uncertainty}
  \Pi_i: \quad G_p(s) = G(s)\big(1 + w_I(s)\Delta_I(s)\big); \quad \abs{\Delta_I(j\w)} \le 1 \ \forall\w
 \end{equation}
 and \(w_I\) should be chosen such that all possible plants \(G^\prime\) are contained in the set \(\Pi_i\).
 Using input multiplicative uncertainty, robust stability is equivalent to have \cite{skogestad07_multiv_feedb_contr}:
 \begin{align*}
  \text{RS} \Leftrightarrow & |w_I T| \le 1 \quad \forall G^\prime \in \Pi_I, \ \forall\omega \\
  \Leftrightarrow & \left| w_I \frac{G^\prime K H_L}{1 + G^\prime K H_L} \right| \le 1 \quad \forall G^\prime \in \Pi_I ,\ \forall\omega \\
  \Leftrightarrow & \left| w_I \frac{G^\prime G^{-1} {H_H}^{-1} H_L}{1 + G^\prime G^{-1} {H_H}^{-1} H_L} \right| \le 1 \quad \forall G^\prime \in \Pi_I ,\ \forall\omega \\
  \Leftrightarrow & \left| w_I \frac{(1 + w_I \Delta) {H_H}^{-1} H_L}{1 + (1 + w_I \Delta) {H_H}^{-1} H_L} \right| \le 1 \quad \forall \Delta, \ |\Delta| \le 1 ,\ \forall\omega \\
  \Leftrightarrow & \left| w_I \frac{(1 + w_I \Delta) H_L}{1 + w_I \Delta H_L} \right| \le 1 \quad \forall \Delta, \ |\Delta| \le 1 ,\ \forall\omega \\
  \Leftrightarrow & \left| H_L w_I \right| \frac{1 + |w_I|}{1 - |w_I H_L|} \le 1, \quad 1 - |w_I H_L| > 0 \quad \forall\omega \\
  \Leftrightarrow & \left| H_L w_I \right| (2 + |w_I|) \le 1, \quad 1 - |w_I H_L| > 0 \quad \forall\omega \\
  \Leftrightarrow & \left| H_L w_I \right| (2 + |w_I|) \le 1 \quad \forall\omega
 \end{align*}
 Robust stability is then guaranteed by having the low pass filter \(H_L\) satisfying \eqref{eq:detail_control_robust_stability}.
 \begin{equation}\label{eq:detail_control_robust_stability}
  \text{RS} \Leftrightarrow |H_L| \le \frac{1}{|w_I| (2 + |w_I|)}\quad \forall \omega
 \end{equation}
 To ensure robust stability condition \eqref{eq:detail_control_nominal_perf_hl} can be used if \(w_L\) is designed in such a way that \(|w_L| \ge |w_I| (2 + |w_I|)\).\\
 \paragraph{Robust Performance (RP)}
 \label{sec:org73ddee1}
 Robust performance is a property for a controlled system to have its performance guaranteed even though the dynamics of the plant is changing within specified bounds.
 For robust performance, we then require to have the performance condition valid for all possible plants in the defined uncertainty set:
 \begin{subnumcases}{\text{RP} \Leftrightarrow}
  |w_H S| \le 1 \quad \forall G^\prime \in \Pi_I, \ \forall\omega \label{eq:detail_control_robust_perf_S}\\
  |w_L T| \le 1 \quad \forall G^\prime \in \Pi_I, \ \forall\omega \label{eq:detail_control_robust_perf_T}
 \end{subnumcases}
 Let's transform condition \eqref{eq:detail_control_robust_perf_S} into a condition on the complementary filters
 \begin{align*}
  & \left| w_H S \right| \le 1 \quad \forall G^\prime \in \Pi_I, \ \forall\omega \\
  \Leftrightarrow & \left| w_H \frac{1}{1 + G^\prime G^{-1} H_H^{-1} H_L} \right| \le 1 \quad \forall G^\prime \in \Pi_I, \ \forall\omega \\
  \Leftrightarrow & \left| \frac{w_H H_H}{1 + \Delta w_I H_L} \right| \le 1 \quad \forall \Delta, \ |\Delta| \le 1, \ \forall\omega \\
  \Leftrightarrow & \frac{|w_H H_H|}{1 - |w_I H_L|} \le 1, \ \forall\omega \\
  \Leftrightarrow & | w_H H_H | + | w_I H_L | \le 1, \ \forall\omega \\
 \end{align*}
 The same can be done with condition \eqref{eq:detail_control_robust_perf_T}
 \begin{align*}
  & \left| w_L T \right| \le 1 \quad \forall G^\prime \in \Pi_I, \ \forall\omega \\
  \Leftrightarrow & \left| w_L \frac{G^\prime G^{-1} H_H^{-1} H_L}{1 + G^\prime G^{-1} H_H^{-1} H_L} \right| \le 1 \quad \forall G^\prime \in \Pi_I, \ \forall\omega \\
  \Leftrightarrow & \left| w_L H_L \frac{1 + w_I \Delta}{1 + w_I \Delta H_L} \right| \le 1 \quad \forall \Delta, \ |\Delta| \le 1, \ \forall\omega \\
  \Leftrightarrow & \left| w_L H_L \right| \frac{1 + |w_I|}{1 - |w_I H_L|} \le 1 \quad \forall\omega \\
  \Leftrightarrow & \left| H_L \right| \le \frac{1}{|w_L| (1 + |w_I|) + |w_I|} \quad \forall\omega \\
 \end{align*}
 Robust performance is then guaranteed if \eqref{eq:detail_control_robust_perf_a} and \eqref{eq:detail_control_robust_perf_b} are satisfied.
 \begin{subnumcases}\label{eq:detail_control_robust_performance}
 {\text{RP} \Leftrightarrow}
  | w_H H_H | + | w_I H_L | \le 1, \ \forall\omega \label{eq:detail_control_robust_perf_a}\\
  \left| H_L \right| \le \frac{1}{|w_L| (1 + |w_I|) + |w_I|} \quad \forall\omega \label{eq:detail_control_robust_perf_b}
 \end{subnumcases}
 One should be aware than when looking for a robust performance condition, only the worst case is evaluated and using the robust stability condition may lead to conservative control.
 \section{Analytical formulas for complementary filters?}
 \label{sec:org23bab6a}
 \label{ssec:detail_control_analytical_complementary_filters}
 \section{Numerical Example}
 \label{sec:org082409f}
 \label{ssec:detail_control_simulations}
 \paragraph{Procedure}
 \label{sec:orgebbfce4}
 In order to apply this control technique, we propose the following procedure:
 \begin{enumerate}
 \item Identify the plant to be controlled in order to obtain \(G\)
 \item Design the weighting function \(w_I\) such that all possible plants \(G^\prime\) are contained in the set \(\Pi_i\)
 \item Translate the performance requirements into upper bounds on the complementary filters (as explained in Sec. \ref{ssec:detail_control_trans_perf})
 \item Design the weighting functions \(w_H\) and \(w_L\) and generate the complementary filters using \(\hinf\text{-synthesis}\) (as further explained in Sec. \ref{ssec:detail_control_hinf_method}).
 If the synthesis fails to give filters satisfying the upper bounds previously defined, either the requirements have to be reworked or a better model \(G\) that will permits to have a smaller \(w_I\) should be obtained.
 If one does not want to use the \(\mathcal{H}_\infty\) synthesis, one can use pre-made complementary filters given in Sec. \ref{ssec:detail_control_analytical_complementary_filters}.
 \item If \(K = \left( G H_H \right)^{-1}\) is not proper, a low pass filter should be added
 \item Design a pre-filter \(K_r\) if requirements on input usage or response to reference change are not met
 \item Control implementation: Filter the measurement with \(H_L\), implement the controller \(K\) and the pre-filter \(K_r\) as shown on Fig. \ref{fig:detail_control_sf_arch_class_prefilter}
 \end{enumerate}
 \paragraph{Plant}
 \label{sec:orgfc350fc}
 Let's consider the problem of controlling an active vibration isolation system that consist of a mass \(m\) to be isolated, a piezoelectric actuator and a geophone.
 We represent this system by a mass-spring-damper system as shown Fig. \ref{fig:detail_control_mech_sys_alone} where \(m\) typically represents the mass of the payload to be isolated, \(k\) and \(c\) represent respectively the stiffness and damping of the mount.
 \(w\) is the ground motion.
 The values for the parameters of the models are
 \[ m = \SI{20}{\kg}; \quad k = 10^4\si{\N/\m}; \quad c = 10^2\si{\N\per(\m\per\s)} \]
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1]{figs/detail_control_mech_sys_alone.png}
 \caption{\label{fig:detail_control_mech_sys_alone}Model of the positioning system}
 \end{figure}
 The model of the plant \(G(s)\) from actuator force \(F\) to displacement \(x\) is then
 \begin{equation}
 G(s) = \frac{1}{m s^2 + c s + k}
 \end{equation}
 Its bode plot is shown on Fig. \ref{fig:detail_control_bode_plot_mech_sys}.
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1]{figs/detail_control_bode_plot_mech_sys.png}
 \caption{\label{fig:detail_control_bode_plot_mech_sys}Bode plot of the transfer function \(G(s)\) from \(F\) to \(x\)}
 \end{figure}
 \paragraph{Requirements}
 \label{sec:orgd766e8b}
 The control objective is to isolate the displacement \(x\) of the mass from the ground motion \(w\).
 The disturbance rejection should be at least \(10\) at \(\SI{2}{\hertz}\) and with a slope of \(-2\) below \(\SI{2}{\hertz}\) until a rejection of \(10^4\).
 Closed-loop bandwidth should be less than \(\SI{20}{\hertz}\) (because of time delay induced by limited sampling frequency?).
 Noise attenuation should be at least \(10\) above \(\SI{40}{\hertz}\) and \(100\) above \(\SI{500}{\hertz}\)
 Robustness to unmodelled dynamics.
 We model the uncertainty on the dynamics of the plant by a multiplicative weight
 \begin{equation}
  w_I(s) = \frac{\tau s + r_0}{(\tau/r_\infty) s + 1}
 \end{equation}
 where \(r_0=0.1\) is the relative uncertainty at steady-state, \(1/\tau=\SI{100}{\hertz}\) is the frequency at which the relative uncertainty reaches \(\SI{100}{\percent}\), and \(r_\infty=10\) is the magnitude of the weight at high frequency.
 All the requirements on \(H_L\) and \(H_H\) are represented on Fig. \ref{fig:detail_control_spec_S_T}.
 \begin{itemize}
 \item[{$\square$}] TODO: Make Matlab code to plot the specifications
 \end{itemize}
 \begin{figure}[htbp]
 \begin{subfigure}{0.49\textwidth}
 \begin{center}
 \includegraphics[scale=1,width=0.95\linewidth]{figs/detail_control_spec_S_T.png}
 \end{center}
 \subcaption{\label{fig:detail_control_spec_S_T}Closed loop specifications}
 \end{subfigure}
 \begin{subfigure}{0.49\textwidth}
 \begin{center}
 \includegraphics[scale=1,width=0.95\linewidth]{figs/detail_control_hinf_filters_result_weights.png}
 \end{center}
 \subcaption{\label{fig:detail_control_hinf_filters_result_weights}Obtained complementary filters}
 \end{subfigure}
 \caption{\label{fig:detail_control_spec_S_T_obtained_filters}Caption with reference to sub figure (\subref{fig:detail_control_spec_S_T}) (\subref{detail_control_hinf_filters_result_weights})}
 \end{figure}
 \paragraph{Design of the filters}
 \label{sec:org11f2da3}
 \textbf{Or maybe use analytical formulas as proposed here: \href{file:///home/thomas/Cloud/research/papers/dehaeze20\_virtu\_senso\_fusio/matlab/index.org}{Complementary filters using analytical formula}}
 We then design \(w_L\) and \(w_H\) such that their magnitude are below the upper bounds shown on Fig. \ref{fig:detail_control_hinf_filters_result_weights}.
 \begin{subequations}
  \begin{align}
    w_L &= \frac{(s+22.36)^2}{0.005(s+1000)^2}\\
    w_H &= \frac{1}{0.0005(s+0.4472)^2}
  \end{align}
 \end{subequations}
 After the \(\hinf\text{-synthesis}\), we obtain \(H_L\) and \(H_H\), and we plot their magnitude on phase on Fig. \ref{fig:detail_control_hinf_filters_result_weights}.
 \begin{subequations}
  \begin{align}
    H_L &= \frac{0.0063957 (s+1016) (s+985.4) (s+26.99)}{(s+57.99) (s^2 + 65.77s + 2981)}\\
    H_H &= \frac{0.9936 (s+111.1) (s^2 + 0.3988s + 0.08464)}{(s+57.99) (s^2 + 65.77s + 2981)}
  \end{align}
 \end{subequations}
 \paragraph{Controller analysis}
 \label{sec:orgbd9d52a}
 The controller is \(K = \left( H_H G \right)^{-1}\).
 A low pass filter is added to \(K\) so that it is proper and implementable.
 The obtained controller is shown on Fig. \ref{fig:detail_control_bode_Kfb}.
 It is implemented as shown on Fig. \ref{fig:detail_control_mech_sys_alone_ctrl}.
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1]{figs/detail_control_mech_sys_alone_ctrl.png}
 \caption{\label{fig:detail_control_mech_sys_alone_ctrl}Control of a positioning system}
 \end{figure}
 \begin{figure}[htbp]
 \begin{subfigure}{0.49\textwidth}
 \begin{center}
 \includegraphics[scale=1,width=0.95\linewidth]{figs/detail_control_bode_Kfb.png}
 \end{center}
 \subcaption{\label{fig:detail_control_bode_Kfb}Controller $K$}
 \end{subfigure}
 \begin{subfigure}{0.49\textwidth}
 \begin{center}
 \includegraphics[scale=1,width=0.95\linewidth]{figs/detail_control_bode_plot_loop_gain_robustness.png}
 \end{center}
 \subcaption{\label{fig:detail_control_bode_plot_loop_gain_robustness}Loop Gain}
 \end{subfigure}
 \caption{\label{fig:detail_control_bode_Kfb_loop_gain}Caption with reference to sub figure (\subref{fig:detail_control_bode_Kfb}) (\subref{fig:detail_control_bode_plot_loop_gain_robustness})}
 \end{figure}
 \paragraph{Robustness analysis}
 \label{sec:org85bb3dc}
 The robust stability can be access on the nyquist plot (Fig. \ref{fig:detail_control_nyquist_robustness}).
 The robust performance is shown on Fig. \ref{fig:detail_control_robust_perf}.
 \begin{figure}[htbp]
 \begin{subfigure}{0.49\textwidth}
 \begin{center}
 \includegraphics[scale=1,scale=0.8]{figs/detail_control_nyquist_robustness.png}
 \end{center}
 \subcaption{\label{fig:detail_control_nyquist_robustness}Robust Stability}
 \end{subfigure}
 \begin{subfigure}{0.49\textwidth}
 \begin{center}
 \includegraphics[scale=1,scale=0.8]{figs/detail_control_robust_perf.png}
 \end{center}
 \subcaption{\label{fig:detail_control_robust_perf}Robust performance}
 \end{subfigure}
 \caption{\label{fig:fig_label}Caption with reference to sub figure (\subref{fig:detail_control_nyquist_robustness}) (\subref{fig:detail_control_robust_perf})}
 \end{figure}
 \section{Experimental Validation?}
 \label{sec:org8a7211e}
 \label{ssec:detail_control_exp_validation}
 \href{file:///home/thomas/Cloud/research/papers/dehaeze20\_virtu\_senso\_fusio/matlab/index.org}{Experimental Validation}
 \section*{Conclusion}
 \label{sec:org40e7289}
 \begin{itemize}
 \item[{$\square$}] Discuss how useful it is as the bandwidth can be changed in real time with analytical formulas of second order complementary filters.
 Maybe make a section about that.
 Maybe give analytical formulas of second order complementary filters in the digital domain?
 \item[{$\square$}] Say that it will be validated with the nano-hexapod
 \item[{$\square$}] Disadvantages:
 \begin{itemize}
 \item not optimal
 \item computationally intensive?
 \item lead to inverse control which may not be wanted in many cases. Add reference.
 \end{itemize}
 \end{itemize}
 \chapter*{Conclusion}
 \label{sec:orgcbb5ce3}
 \label{sec:detail_control_conclusion}
 \printbibliography[heading=bibintoc,title={Bibliography}]
 \end{document}