#+TITLE: Control Optimization
:DRAWER:
#+LANGUAGE: en
#+EMAIL: dehaeze.thomas@gmail.com
#+AUTHOR: Dehaeze Thomas

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* Build                                                             :noexport:
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* Notes                                                             :noexport:
** Notes

Prefix is =detail_control=

Up to now, basic control strategy ("decentralized" control in the frame of the struts), HAC-LAC

Discussion about:
- *Use multiple sensors*: Discussion about two sensor control VS HAC-LAC VS Sensor Fusion ?
- *Decoupling strategies*
  - [ ] file:~/Cloud/research/matlab/decoupling-strategies/svd-control.org
- *Control optimization*

** Unused
*** Table with multi sensor control

#+name: tab:detail_control_sensor_review
#+caption: Review of Stewart platforms integrating multiple sensors
#+attr_latex: :environment tabularx :width 0.9\linewidth :align Xcccc
#+attr_latex: :center t :booktabs t :font \scriptsize
|   | *Actuators*      | *Sensors*                                         | *Control*                                                                             | Main Object                                                                                                      | *Reference*                                                                                                            |
|---+------------------+---------------------------------------------------+---------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------|
| X | Magnetostrictive | Force (collocated), Accelerometers                | Two layers: Decentralized IFF, Robust Adaptive Control                                | Two layer control for active damping and vibration isolation                                                     | [[cite:&geng95_intel_contr_system_multip_degree]]                                                                          |
| X | Voice Coil       | Accelerometer (collocated), ext. Rx/Ry sensors    | Cartesian acceleration feedback (isolation) + 2DoF pointing control (external sensor) | Decoupling, both vibration + pointing control                                                                    | [[cite:&li01_simul_vibrat_isolat_point_contr]]                                                                             |
|   | Voice Coil       | Geophone + Eddy Current (Struts, collocated)      | Decentralized (Sky Hook) + Centralized (modal) Control                                |                                                                                                                  | [[cite:&pu11_six_degree_of_freed_activ]]                                                                                   |
| X | Piezoelectric    | Force Sensor + Accelerometer                      | HAC-LAC (IFF + FxLMS)                                                                 | Dynamic Model + Vibration Control                                                                                | [[cite:&wang16_inves_activ_vibrat_isolat_stewar]]                                                                          |
| X | Voice Coil       | Force sensors (strus) + accelerometer (cartesian) | Decentralized Force Feedback + Centralized H2 control based on accelerometers         |                                                                                                                  | [[cite:&xie17_model_contr_hybrid_passiv_activ]]                                                                            |
|---+------------------+---------------------------------------------------+---------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------|
| X | Voice Coil       | Force (HF) and Inertial (LF)                      | *Sensor Fusion*, *Two Sensor Control*                                                 |                                                                                                                  | [[cite:&tjepkema12_activ_ph;&tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip]]                                   |
| X | Voice Coil       | Force (HF) and Inertial (LF)                      | *Sensor Fusion*, LQG, Decentralized                                                   | Combine force/inertial sensors. Comparison of force sensor and inertial sensors. Issue on non-minimum phase zero | [[cite:&hauge04_sensor_contr_space_based_six]]                                                                             |
| X | Voice Coil       | Force, LVDT, Geophones                            | *two sensor control* LQG, Force + geophones for vibration, LVDT for pointing          | Centralized control is no better than decentralized. Geophone + Force MISO control is good                       | cite:&thayer98_stewar;&thayer02_six_axis_vibrat_isolat_system                                                          |
|---+------------------+---------------------------------------------------+---------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------|
| X | Piezoelectric    | Force, Position                                   | Vibration isolation, Model-Based, Modal control: 6x PI controllers                    | Stiffness of flexible joints is compensated using feedback, then the system is decoupled in the modal space      | [[cite:&yang19_dynam_model_decoup_contr_flexib]]                                                                           |
|   | Voice Coil       | Force                                             | IFF, centralized (decouple) + decentralized (coupled)                                 | Specific geometry: decoupled force plant. Better perf with centralized IFF                                       | [[cite:&mcinroy99_dynam;&mcinroy99_precis_fault_toler_point_using_stewar_platf;&mcinroy00_desig_contr_flexur_joint_hexap]] |

*** "Closed-Loop" complementary filters
<<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 Figure 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]].

#+begin_src latex :file detail_control_feedback_sensor_fusion.pdf
\tikzset{block/.default={0.8cm}{0.8cm}}
\tikzset{addb/.append style={scale=0.7}}
\tikzset{node distance=0.6}
\begin{tikzpicture}
  \node[addb={+}{}{}{}{-}] (addfb) at (0, 0){};
  \node[block, right=1 of addfb] (L){$L(s)$};
  \node[addb={+}{}{}{}{}, right=1 of L] (adddy){};

  \draw[<-] (addfb.west) -- ++(-1, 0) node[above right]{$\hat{x}_2$};
  \draw[->] (addfb.east) -- (L.west);
  \draw[->] (L.east) -- (adddy.west);
  \draw[->] (adddy.east) -- ++(1.4, 0) node[above left]{$\hat{x}$};
  \draw[->] ($(adddy.east) + (0.5, 0)$) node[branch]{} -- ++(0, -0.8) coordinate(botc) -| (addfb.south);
  \draw[<-] (adddy.north) -- ++(0, 1) node[below right]{$\hat{x}_1$};

  \begin{scope}[on background layer]
    \node[fit={(L.north-|addfb.west) (botc)}, fill=black!10!white, draw, inner sep=6pt] (supersensor) {};
  \end{scope}
\end{tikzpicture}
#+end_src

#+name: fig:detail_control_feedback_sensor_fusion
#+caption: "Closed-Loop" complementary filters.
#+RESULTS:
[[file:figs/detail_control_feedback_sensor_fusion.png]]

Consider the feedback architecture of Figure ref:fig:detail_control_feedback_sensor_fusion, with two inputs $\hat{x}_1$ and $\hat{x}_2$, and one output $\hat{x}$.
The output $\hat{x}$ is linked to the inputs by eqref:eq:detail_control_closed_loop_complementary_filters.

\begin{equation}\label{eq:detail_control_closed_loop_complementary_filters}
\hat{x} = \underbrace{\frac{1}{1 + L(s)}}_{S(s)} \hat{x}_1 + \underbrace{\frac{L(s)}{1 + L(s)}}_{T(s)} \hat{x}_2
\end{equation}

As for any classical feedback architecture, we have that the sum of the sensitivity transfer function $S(s)$ and complementary sensitivity transfer function $T_(s)$ is equal to one eqref:eq:detail_control_sensitivity_sum.

\begin{equation}\label{eq:detail_control_sensitivity_sum}
S(s) + T(s) = 1
\end{equation}

Therefore, provided that the the closed-loop system in Figure ref:fig:detail_control_feedback_sensor_fusion is stable, it can be used as a set of two complementary filters.
Two sensors can then be merged as shown in Figure ref:fig:detail_control_feedback_sensor_fusion_arch.

#+begin_src latex :file detail_control_feedback_sensor_fusion_arch.pdf
\tikzset{block/.default={0.8cm}{0.8cm}}
\tikzset{addb/.append style={scale=0.7}}
\tikzset{node distance=0.6}
\begin{tikzpicture}
  \node[addb={+}{}{}{}{-}] (addfb) at (0, 0){};
  \node[block, right=1 of addfb] (L){$L(s)$};
  \node[addb={+}{}{}{}{}, right=1 of L] (adddy){};

  \node[block, left=1.2 of addfb]    (sensor2){Sensor 2};
  \node[block, above=0.4 of sensor2] (sensor1){Sensor 1};
  \node[branch, left=0.6 of sensor2] (x){};

  \draw[->] (addfb.east) -- (L.west);
  \draw[->] (L.east) -- (adddy.west);
  \draw[->] (adddy.east) -- ++(1.4, 0) node[above left]{$\hat{x}$};
  \draw[->] ($(adddy.east) + (0.5, 0)$) node[branch]{} -- ++(0, -0.8) coordinate(botc) -| (addfb.south);
  \draw[->] (x.center) |- (sensor1.west);
  \draw[->] ($(x)-(0.8,0)$) node[above right]{$x$} -- (sensor2.west);
  \draw[->] (sensor2.east)node[above right=0 and 0.25]{$\hat{x}_2$} -- (addfb.west);
  \draw[->] (sensor1.east)node[above right=0 and 0.25]{$\hat{x}_1$} -| (adddy.north);

  \begin{scope}[on background layer]
    \node[fit={(x|-sensor1.north) (botc)}, fill=black!10!white, draw, inner sep=9pt] (supersensor) {};
    \node[fit={(sensor1.north-|addfb.west) (botc)}, fill=black!20!white, draw, inner sep=6pt] (feedbackfilter) {};
    \node[fit={(sensor2.west|-botc) (sensor1.north east)}, fill=black!20!white, draw, inner sep=6pt] (sensors) {};
    \node[above, align=center] at (sensors.south) {{\tiny Normalized}\\[-0.5em]{\tiny sensors}};
    \node[below, align=center] at (feedbackfilter.north) {{\tiny "Closed-Loop"}\\[-0.5em]{\tiny complementary filters}};
  \end{scope}
\end{tikzpicture}
#+end_src

#+name: fig:detail_control_feedback_sensor_fusion_arch
#+caption: Classical feedback architecture used for sensor fusion.
#+RESULTS:
[[file:figs/detail_control_feedback_sensor_fusion_arch.png]]

One of the main advantage of implementing and designing complementary filters using the feedback architecture of Figure ref:fig:detail_control_feedback_sensor_fusion is that all the tools of the linear control theory can be applied for the design of the filters.
If one want to shape both $\frac{\hat{x}}{\hat{x}_1}(s) = S(s)$ and $\frac{\hat{x}}{\hat{x}_2}(s) = T(s)$, the $\mathcal{H}_\infty$ mixed-sensitivity synthesis can be easily applied.

To do so, weighting functions $W_1(s)$ and $W_2(s)$ are added to respectively shape $S(s)$ and $T(s)$ (Figure ref:fig:detail_control_feedback_synthesis_architecture).
Then the system is rearranged to form the generalized plant $P_L(s)$ shown in Figure ref:fig:detail_control_feedback_synthesis_architecture_generalized_plant.
The $\mathcal{H}_\infty$ mixed-sensitivity synthesis can finally be performed by applying the standard $\mathcal{H}_\infty$ synthesis to the generalized plant $P_L(s)$ which is described by eqref:eq:detail_control_generalized_plant_mixed_sensitivity.

\begin{equation}\label{eq:detail_control_generalized_plant_mixed_sensitivity}
  \begin{bmatrix} z \\ v \end{bmatrix} = P_L(s) \begin{bmatrix} w_1 \\ w_2 \\ u \end{bmatrix}; \quad P_L(s) = \begin{bmatrix}
    \phantom{+}W_1(s) & 0      & \phantom{+}1 \\
              -W_1(s) & W_2(s) &           -1
  \end{bmatrix}
\end{equation}

The output of the synthesis is a filter $L(s)$ such that the "closed-loop" $\mathcal{H}_\infty$ norm from $[w_1,\ w_2]$ to $z$ of the system in Figure ref:fig:detail_control_feedback_sensor_fusion is less than one eqref:eq:detail_control_comp_filters_feedback_obj.

\begin{equation}\label{eq:detail_control_comp_filters_feedback_obj}
  \left\| \begin{matrix} \frac{z}{w_1} \\ \frac{z}{w_2} \end{matrix} \right\|_\infty = \left\| \begin{matrix} \frac{1}{1 + L(s)} W_1(s) \\ \frac{L(s)}{1 + L(s)} W_2(s) \end{matrix} \right\|_\infty \le 1
\end{equation}

If the synthesis is succescful, the transfer functions from $\hat{x}_1$ to $\hat{x}$ and from $\hat{x}_2$ to $\hat{x}$ have their magnitude bounded by the inverse magnitude of the corresponding weighting functions.
The sensor fusion can then be implemented using the feedback architecture in Figure ref:fig:detail_control_feedback_sensor_fusion_arch or more classically as shown in Figure ref:fig:detail_control_sensor_fusion_overview by defining the two complementary filters using eqref:eq:detail_control_comp_filters_feedback.
The two architectures are equivalent regarding their inputs/outputs relationships.

\begin{equation}\label{eq:detail_control_comp_filters_feedback}
  H_1(s) = \frac{1}{1 + L(s)}; \quad H_2(s) = \frac{L(s)}{1 + L(s)}
\end{equation}

#+begin_src latex :file detail_control_feedback_synthesis_architecture.pdf
\tikzset{block/.default={0.8cm}{0.8cm}}
\tikzset{addb/.append style={scale=0.7}}
\tikzset{node distance=0.6}
\begin{tikzpicture}
  \node[block] (W2) at (0,0) {$W_2(s)$};
  \node[addb={+}{}{}{}{-}, right=0.8 of W2] (addfb){};
  \node[addb={+}{}{}{}{}, right=4.5 of W2] (adddy){};
  \node[block, above=0.8 of adddy] (W1){$W_1(s)$};

  \draw[<-] (W2.west) -- ++(-0.8, 0) node[above right]{$w_2$};
  \draw[->] (W2.east) -- (addfb.west) node[above left]{$\tilde{w}_2$};
  \draw[->] (addfb.east) -- ++(1, 0) node[above left]{$v$};
  \draw[<-] (adddy.west) -- ++(-1, 0) node[above right]{$u$};
  \draw[->] (adddy.east) -- ++(1.4, 0) node[above left]{$z$};
  \draw[->] (W1.south) -- (adddy.north) node[above right]{$\tilde{w}_1$};
  \draw[<-] (W1.north) -- ++(0, 0.8) node[below right]{$w_1$};
  \draw[->] ($(adddy.east) + (0.5, 0)$) node[branch]{} -- ++(0, -0.8) -| (addfb.south);
\end{tikzpicture}
#+end_src

#+begin_src latex :file detail_control_feedback_synthesis_architecture_generalized_plant.pdf
\tikzset{block/.default={0.8cm}{0.8cm}}
\tikzset{addb/.append style={scale=0.7}}
\tikzset{node distance=0.6}
\begin{tikzpicture}
  \node[block={4.5cm}{3.0cm}, fill=black!10!white] (P) {};
  \node[above] at (P.north) {$P_L(s)$};

  \coordinate[] (inputw1) at ($(P.south west)!0.75!(P.north west) + (-0.7, 0)$);
  \coordinate[] (inputw2) at ($(P.south west)!0.40!(P.north west) + (-0.7, 0)$);
  \coordinate[] (inputu)  at ($(P.south west)!0.15!(P.north west) + (-0.7, 0)$);

  \coordinate[] (outputz) at ($(P.south east)!0.75!(P.north east) + ( 0.7, 0)$);
  \coordinate[] (outputv) at ($(P.south east)!0.40!(P.north east) + ( 0.7, 0)$);

  \node[block, right=1.2 of inputw2] (W2){$W_2(s)$};
  \node[block, right=1.2 of inputw1] (W1){$W_1(s)$};
  \node[addb={+}{}{}{}{}, right=0.8 of W1] (add) {};
  \node[addb={+}{}{-}{}{},  right=1.8 of W2] (sub) {};

  \draw[->] (inputw2) node[above right]{$w_2$} -- (W2.west);
  \draw[->] (inputw1) node[above right]{$w_1$} -- (W1.west);
  \draw[->] (inputu)  node[above right]{$u$}   -| (add.south);
  \draw[->] (W2.east) -- (sub.west);
  \draw[->] (W1.east) -- (add.west);
  \draw[->] (add.east) -- (outputz)node[above left]{$z$};
  \draw[->] (sub.east) -- (outputv)node[above left]{$v$};
  \draw[->] (add-|sub) node[branch]{} -- (sub.north);
\end{tikzpicture}
#+end_src

#+name: fig:detail_control_h_inf_mixed_sensitivity_synthesis
#+caption: $\mathcal{H}_\infty$ mixed-sensitivity synthesis
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:detail_control_feedback_synthesis_architecture}Feedback architecture with included weights}
#+attr_latex: :options {0.58\textwidth}
#+begin_subfigure
#+attr_latex: :scale 1
[[file:figs/detail_control_feedback_synthesis_architecture.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_control_feedback_synthesis_architecture_generalized_plant}Generalized plant}
#+attr_latex: :options {0.38\textwidth}
#+begin_subfigure
#+attr_latex: :scale 1
[[file:figs/detail_control_feedback_synthesis_architecture_generalized_plant.png]]
#+end_subfigure
#+end_figure

As an example, two "closed-loop" complementary filters are designed using the $\mathcal{H}_\infty$ mixed-sensitivity synthesis.
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 Figure 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 Figure ref:fig:detail_control_hinf_filters_results_mixed_sensitivity confirming that the synthesis is succescful.
The obtained "closed-loop" complementary filters are indeed equal to the ones obtained in Section ref:ssec:detail_control_hinf_method.

#+begin_src matlab
%% Design of "Closed-loop" complementary filters
% Design of the Weighting Functions
W1 = generateWF('n', 3, 'w0', 2*pi*10, 'G0', 1000, 'Ginf', 1/10, 'Gc', 0.45);
W2 = generateWF('n', 2, 'w0', 2*pi*10, 'G0', 1/10, 'Ginf', 1000, 'Gc', 0.45);

% Generalized plant for "closed-loop" complementary filter synthesis
P = [ W1 0   1;
     -W1 W2 -1];

% Standard H-Infinity Synthesis
[L, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');

% Complementary filters
H1 = inv(1 + L);
H2 = 1 - H1;
#+end_src

#+begin_src matlab :exports none :results none
%% Bode plot of the obtained complementary filters after H-infinity mixed-sensitivity synthesis
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');

% Magnitude
ax1 = nexttile([2, 1]);
hold on;
set(gca,'ColorOrderIndex',1)
plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$|W_1|^{-1}$');
set(gca,'ColorOrderIndex',2)
plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$|W_2|^{-1}$');

set(gca,'ColorOrderIndex',1)
plot(freqs, abs(squeeze(freqresp(H1, freqs, 'Hz'))), '-', 'DisplayName', '$H_1$');
set(gca,'ColorOrderIndex',2)
plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), '-', 'DisplayName', '$H_2$');

plot(freqs, abs(squeeze(freqresp(L, freqs, 'Hz'))), 'k--', 'DisplayName', '$|L|$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]); ylabel('Magnitude');
ylim([1e-3, 1e3]);
yticks([1e-3, 1e-2, 1e-1, 1, 1e1, 1e2, 1e3]);
yticklabels({'', '$10^{-2}$', '', '$10^0$', '', '$10^2$', ''});
leg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 3);
leg.ItemTokenSize(1) = 18;

% Phase
ax2 = nexttile;
hold on;
set(gca,'ColorOrderIndex',1)
plot(freqs, 180/pi*phase(squeeze(freqresp(H1, freqs, 'Hz'))), '-');
set(gca,'ColorOrderIndex',2)
plot(freqs, 180/pi*phase(squeeze(freqresp(H2, freqs, 'Hz'))), '-');
hold off;
set(gca, 'XScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
yticks([-180:90:180]);
ylim([-180, 200])
yticklabels({'-180', '', '0', '', '180'})

linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/detail_control_hinf_filters_results_mixed_sensitivity.pdf', 'width', 'wide', 'height', 600);
#+end_src

#+name: fig:detail_control_hinf_filters_results_mixed_sensitivity
#+caption: Bode plot of the obtained complementary filters after $\mathcal{H}_\infty$ mixed-sensitivity synthesis
#+RESULTS:
[[file:figs/detail_control_hinf_filters_results_mixed_sensitivity.png]]

*** Analytical formulas for test model

#+begin_src matlab
%% Analytical Formula for the Jacobian Matrix
% Create symbolic variables for all parameters
syms l h la ha m I real
syms c1 c2 c3 real
syms k1 k2 k3 real

% Unit vectors of the actuators (symbolic)
s1 = [1; 0];  % Actuator 1 direction (horizontal)
s2 = [0; 1];  % Actuator 2 direction (vertical)
s3 = [0; 1];  % Actuator 3 direction (vertical)

% Location of the joints with respect to the center of mass (symbolic)
Mb1 = [-l/2; -ha];    % Joint 1 position vector
Mb2 = [-la;  -h/2];   % Joint 2 position vector
Mb3 = [ la;  -h/2];   % Joint 3 position vector

% Jacobian matrix (Center of Mass)
J_CoM = [s1', Mb1(1)*s1(2) - Mb1(2)*s1(1);
         s2', Mb2(1)*s2(2) - Mb2(2)*s2(1);
         s3', Mb3(1)*s3(2) - Mb3(2)*s3(1)];

% Display the symbolic Jacobian matrix
disp('Symbolic Jacobian Matrix (J_CoM):');
disp(J_CoM);

% Jacobian at the Center of stiffness {K}
Mb1 = [-l/2; 0];
Mb2 = [-la; -h/2+ha];
Mb3 = [ la; -h/2+ha];

J_CoK = [s1', Mb1(1)*s1(2) - Mb1(2)*s1(1);
         s2', Mb2(1)*s2(2) - Mb2(2)*s2(1);
         s3', Mb3(1)*s3(2) - Mb3(2)*s3(1)];

% Display the symbolic Jacobian matrix
disp('Symbolic Jacobian Matrix (J_CoK):');
disp(J_CoK);
#+end_src

*** Analytical formulas for modal decoupling

#+begin_src matlab
%% Analytical Formula for the Modal Decoupling
syms l h la ha m I k c s real

% Unit vectors of the actuators
s1 = [1; 0];  % Actuator 1 direction (horizontal)
s2 = [0; 1];  % Actuator 2 direction (vertical)
s3 = [0; 1];  % Actuator 3 direction (vertical)

% Location of the joints with respect to the center of mass (symbolic)
Mb1 = [-l/2; -ha];  % Joint 1 position vector
Mb2 = [-la;  -h/2]; % Joint 2 position vector
Mb3 = [ la;  -h/2]; % Joint 3 position vector

% Calculate the Jacobian matrix (Center of Mass) symbolically
J_CoM = [s1', Mb1(1)*s1(2) - Mb1(2)*s1(1);
         s2', Mb2(1)*s2(2) - Mb2(2)*s2(1);
         s3', Mb3(1)*s3(2) - Mb3(2)*s3(1)];

disp('Symbolic Jacobian Matrix (J_CoM):');
disp(J_CoM);

% Define system matrices
M = diag([m, m, I]);
K_struts = diag([k, k, k]);

% Transform stiffness and damping to Cartesian space
K = J_CoM' * K_struts * J_CoM;

disp('Mass Matrix (M):');
disp(M);

disp('Stiffness Matrix (K):');
disp(K);

MK = simplify(M\K);
disp('M\K Matrix:');
disp(MK);

[V, ~] = eig(MK);
disp(simplify(V));
disp('1st eigen vector:');
disp(simplify(V(:,1)));
disp('2nd eigen vector:');
disp(simplify(V(:,2)));
disp('3rd eigen vector:');
disp(simplify(V(:,3)));

Vinv = inv(V);
disp(simplify(Vinv));
#+end_src

*** Simscape plant

Verify that the collocated plant is the same as the analytical plant.
*Yes they are the same*

#+begin_src matlab
%% Collocated plant
clear io; io_i = 1;
io(io_i) = linio([mdl, '/F1'], 1, 'openinput');  io_i = io_i + 1;
io(io_i) = linio([mdl, '/F2'], 1, 'openinput');  io_i = io_i + 1;
io(io_i) = linio([mdl, '/F3'], 1, 'openinput');  io_i = io_i + 1;
io(io_i) = linio([mdl, '/Actuator_1'], 1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Actuator_2'], 1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Actuator_3'], 1, 'openoutput'); io_i = io_i + 1;

G_L_sim = linearize(mdl, io);
G_L_sim.InputName  = {'F1', 'F2', 'F3'};
G_L_sim.OutputName = {'L1', 'L2', 'L3'};
#+end_src

#+begin_src matlab
%% Accelerometer outputs
% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/F1'], 1, 'openinput');  io_i = io_i + 1;
io(io_i) = linio([mdl, '/F2'], 1, 'openinput');  io_i = io_i + 1;
io(io_i) = linio([mdl, '/F3'], 1, 'openinput');  io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_side'], 1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_side'], 2, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_top'], 1, 'openoutput'); io_i = io_i + 1;
% io(io_i) = linio([mdl, '/Acc_top'], 2, 'openoutput'); io_i = io_i + 1;

G_acc = linearize(mdl, io);
G_acc.InputName  = {'F1', 'F2', 'F3'};
G_acc.OutputName = {'Ax1', 'Ay1', 'Ax2'};
#+end_src

*** Robustness of the decoupling strategies?                      :noexport:
**** Introduction                                                   :ignore:
What happens if we add an additional resonance in the system (Figure ref:fig:model_test_decoupling_spurious_res).

Having less actuator than DoF (under-actuated system):
- modal decoupling: can still control first $n$ modes?
- SVD decoupling: does not matter
- Jacobian decoupling: could give poor decoupling?

#+name: fig:model_test_decoupling_spurious_res
#+caption: Plant with spurious resonance (additional DoF)
[[file:figs/model_test_decoupling_spurious_res.png]]

**** Plant
A multi body model of the system in Figure ref:fig:model_test_decoupling_spurious_res has been made using Simscape.

Its parameters are defined below:
#+begin_src matlab
leq = 20e-3; % Equilibrium length of struts [m]
mr = 5; % [kg]
kr = (2*pi*10)^2*mr; % Stiffness [N/m]
cr = 1e1; % Damping [N/(m/s)]

m = 400 - mr; % Mass [kg]
#+end_src

The plant is then identified and shown in Figure ref:fig:detail_control_coupled_plant_bode_spurious.
The added resonance only slightly modifies the plant around 10Hz.
#+begin_src matlab :exports none
%% Name of the Simulink File
mdl = 'suspended_mass';

%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/F'],  1, 'openinput');  io_i = io_i + 1;
io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1;

Gr = linearize(mdl, io);
Gr.InputName  = {'F1', 'F2', 'F3'};
Gr.OutputName = {'L1', 'L2', 'L3'};
#+end_src

#+begin_src matlab :exports none
figure;
tiledlayout(3, 3, 'TileSpacing', 'Compact', 'Padding', 'None');

for out_i = 1:3
    for in_i = 1:3
        nexttile;
        hold on;
        plot(freqs, abs(squeeze(freqresp(G(out_i,in_i), freqs, 'Hz'))), 'k-', ...
             'DisplayName', sprintf('$\\mathcal{L}_%i/\\tau_%i$', out_i, in_i));
        plot(freqs, abs(squeeze(freqresp(Gr(out_i,in_i), freqs, 'Hz'))), 'k--', ...
             'HandleVisibility', 'off');
        hold off;
        set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
        xlim([1e-1, 2e1]); ylim([1e-6, 1e-2]);
        legend('location', 'northeast', 'FontSize', 8);

        if in_i == 1
            ylabel('Mag. [m/N]')
        else
            set(gca, 'YTickLabel',[]);
        end

        if out_i == 3
            xlabel('Frequency [Hz]')
        else
            set(gca, 'XTickLabel',[]);
        end
    end
end
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/detail_control_coupled_plant_bode_spurious.pdf', 'width', 'full', 'height', 'tall');
#+end_src

#+name: fig:detail_control_coupled_plant_bode_spurious
#+caption: Magnitude of the coupled plant without additional mode (solid) and with the additional mode (dashed).
#+RESULTS:
[[file:figs/detail_control_coupled_plant_bode_spurious.png]]

**** Jacobian Decoupling
The obtained plant is decoupled using the Jacobian matrix.

#+begin_src matlab
Gxr = pinv(J)*Gr*pinv(J');
Gxr.InputName  = {'Fx', 'Fy', 'Mz'};
Gxr.OutputName  = {'Dx', 'Dy', 'Rz'};
#+end_src

The obtained plant is shown in Figure ref:fig:detail_control_jacobian_plant_spurious and is not much different than for the plant without the spurious resonance.
#+begin_src matlab :exports none
figure;

% Magnitude
hold on;
for i_in = 1:3
    for i_out = [i_in+1:3]
        plot(freqs, abs(squeeze(freqresp(Gxr(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
             'HandleVisibility', 'off');
    end
end
plot(freqs, abs(squeeze(freqresp(Gxr(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
     'DisplayName', '$G_{x,r}(i,j)\ i \neq j$');
set(gca,'ColorOrderIndex',1)
for i_in_out = 1:3
    plot(freqs, abs(squeeze(freqresp(Gxr(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_{x,r}(%d,%d)$', i_in_out, i_in_out));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
ylim([1e-7, 1e-1]);
legend('location', 'northeast');
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/detail_control_jacobian_plant_spurious.pdf', 'width', 'wide', 'height', 'normal');
#+end_src

#+name: fig:detail_control_jacobian_plant_spurious
#+caption: Plant with spurious resonance decoupled using the Jacobian matrices $G_{x,r}(s)$
#+RESULTS:
[[file:figs/detail_control_jacobian_plant_spurious.png]]

**** Modal Decoupling
The obtained plant is now decoupled using the modal matrices obtained with the plant not including the added resonance.
#+begin_src matlab
Gmr = inv(Cm)*Gr*inv(Bm);
#+end_src

The obtained decoupled plant is shown in Figure ref:fig:detail_control_modal_plant_spurious.
Compare to the decoupled plant in Figure ref:fig:detail_control_modal_plant, the added resonance induces some coupling, especially around the frequency of the added spurious resonance.
#+begin_src matlab :exports none
figure;

% Magnitude
hold on;
for i_in = 1:3
    for i_out = [i_in+1:3]
        plot(freqs, abs(squeeze(freqresp(Gmr(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
             'HandleVisibility', 'off');
    end
end
plot(freqs, abs(squeeze(freqresp(Gmr(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
     'DisplayName', '$G_{m,r}(i,j)\ i \neq j$');
set(gca,'ColorOrderIndex',1)
for i_in_out = 1:3
    plot(freqs, abs(squeeze(freqresp(Gmr(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_{m,r}(%d,%d)$', i_in_out, i_in_out));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
ylim([1e-6, 1e2]);
legend('location', 'northeast');
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/detail_control_modal_plant_spurious.pdf', 'width', 'wide', 'height', 'normal');
#+end_src

#+name: fig:detail_control_modal_plant_spurious
#+caption: Modal plant including spurious resonance $G_{m,r}(s)$
#+RESULTS:
[[file:figs/detail_control_modal_plant_spurious.png]]

**** SVD Decoupling
The SVD decoupling is performed on the new obtained plant.
The decoupling frequency is slightly shifted in order not to interfere too much with the spurious resonance.

#+begin_src matlab
%% Decoupling frequency [rad/s]
wc = 2*pi*7;

%% System's response at the decoupling frequency
H1 = evalfr(Gr, j*wc);

%% Real approximation of G(j.wc)
D = pinv(real(H1'*H1));
H1 = pinv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));

[U,S,V] = svd(H1);

Gsvdr = inv(U)*Gr*inv(V');
#+end_src

The obtained plant is shown in Figure ref:fig:detail_control_svd_plant_spurious.
#+begin_src matlab :exports none
figure;

% Magnitude
hold on;
for i_in = 1:3
    for i_out = [i_in+1:3]
        plot(freqs, abs(squeeze(freqresp(Gsvdr(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
             'HandleVisibility', 'off');
    end
end
plot(freqs, abs(squeeze(freqresp(Gsvdr(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
     'DisplayName', '$G_{svd,r}(i,j)\ i \neq j$');
set(gca,'ColorOrderIndex',1)
for i_in_out = 1:3
    plot(freqs, abs(squeeze(freqresp(Gsvdr(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_{svd,r}(%d,%d)$', i_in_out, i_in_out));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
ylim([1e-8, 1e-2]);
legend('location', 'northeast');
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/detail_control_svd_plant_spurious.pdf', 'width', 'wide', 'height', 'normal');
#+end_src

#+name: fig:detail_control_svd_plant_spurious
#+caption: SVD decoupled plant including a spurious resonance $G_{svd,r}(s)$
#+RESULTS:
[[file:figs/detail_control_svd_plant_spurious.png]]

*** Analytical formula for complementary filters
#+begin_src matlab
%% 1st order complementary filters
% Declaration of the symbolic variables
syms w0 Ts s z

% Bilinear Transform
s = 2/Ts*(z - 1)/(z + 1);

% First Order Low Pass Filter
Hl = 1/(1 + s/w0);

% Get numerator and denominator
[N,D] = numden(Hl);

% Extract coefficients (from z^0 to z^n)
num = coeffs(N, z);
den = coeffs(D, z);

sprintf('Hl: num = (%s) + (%s) * z^-1', num(2), num(1))
sprintf('Hl: den = (%s) + (%s) * z^-1', den(2), den(1))

% First Order Low Pass Filter
Hh = (s/w0)/(1 + s/w0);

% Get numerator and denominator
[N,D] = numden(Hh);

% Extract coefficients (from z^0 to z^n)
num = coeffs(N, z);
den = coeffs(D, z);

sprintf('Hh: num = (%s) + (%s) * z^-1', num(2), num(1))
sprintf('Hh: den = (%s) + (%s) * z^-1', den(2), den(1))
#+end_src

** DONE [#A] Fix the outline
CLOSED: [2025-04-03 Thu 12:01]

Lot is already discussed in:
- A5 [[file:~/Cloud/work-projects/ID31-NASS/phd-thesis-chapters/A5-simscape-nano-hexapod/simscape-nano-hexapod.org::*Control of Stewart Platforms][Control of Stewart Platforms]]
- Some will be discussed in the geometry optimization (cubic configuration / decoupling / ...)

- [ ] Maybe do not write this section at all?

** DONE [#B] Papers to be included here
CLOSED: [2025-04-03 Thu 12:01]

- [X] H-infinity synthesis of complementary filters:
  [[file:~/Cloud/research/papers/published/dehaeze19_compl_filter_shapin_using_synth/index.org][file:~/Cloud/research/papers/published/dehaeze19_compl_filter_shapin_using_synth/index.org]]
  Or newer version: [[file:~/Cloud/research/papers/dehaeze21_desig_compl_filte/journal/dehaeze21_desig_compl_filte.org][file:~/Cloud/research/papers/dehaeze21_desig_compl_filte/journal/dehaeze21_desig_compl_filte.org]]
- [X] Virtual sensor fusion
  [[file:~/Cloud/research/papers/published/verma19_virtu_senso_fusio_high_preci_contr/elsarticle-template-harv.tex][file:~/Cloud/research/papers/published/verma19_virtu_senso_fusio_high_preci_contr/elsarticle-template-harv.tex]]
  Maybe not this one.
- [X] Feedback control based on complementary filters:
  [[file:~/Cloud/research/papers/dehaeze20_virtu_senso_fusio/index.org][file:~/Cloud/research/papers/dehaeze20_virtu_senso_fusio/index.org]]

** DONE [#A] Copy Paper about Complementary Filter Design
CLOSED: [2025-04-03 Thu 15:25] SCHEDULED: <2025-04-03 Thu>

[[file:~/Cloud/research/papers/dehaeze21_desig_compl_filte/journal/dehaeze21_desig_compl_filte.org][file:~/Cloud/research/papers/dehaeze21_desig_compl_filte/journal/dehaeze21_desig_compl_filte.org]]

- [X] Copy Matlab Code: [[file:~/Cloud/research/papers/dehaeze21_desig_compl_filte/matlab/dehaeze21_desig_compl_filte_matlab.org]]
- [X] Copy Tikz Code: [[file:~/Cloud/research/papers/dehaeze21_desig_compl_filte/tikz/dehaeze21_desig_compl_filte_tikz.org]]
- [X] Rework all labels:
  - [X] sections
  - [X] equations
  - [X] tables

** DONE [#A] Copy paper about closed-loop control with complementary filters
CLOSED: [2025-04-03 Thu 16:30] SCHEDULED: <2025-04-03 Thu>

file:~/Cloud/research/papers/dehaeze20_virtu_senso_fusio/index.org

- [X] Copy Content
- [X] Change citation format
- [X] Copy Tikz figures
- [X] Copy Matlab Code
- [X] Rework all labels
  - [X] sections
  - [X] equations
  - [X] tables

** DONE [#A] Copy paper about Decoupling Control
CLOSED: [2025-04-03 Thu 17:48] SCHEDULED: <2025-04-03 Thu>

file:~/Cloud/research/matlab/decoupling-strategies/svd-control.org
Especially [[file:~/Cloud/research/matlab/decoupling-strategies/svd-control.org::*SVD / Jacobian / Model decoupling comparison][SVD / Jacobian / Model decoupling comparison]]

- [X] Copy Content
- [X] Copy Tikz figures
- [X] Copy Matlab Code
- [X] Rework all labels
  - [X] sections
  - [X] equations
  - [X] tables

** DONE [#A] Verify why SVD decomposition on the proposed example gives such good performance
CLOSED: [2025-04-05 Sat 17:43]

Could be due to symmetry in the system.

** DONE [#B] Rename figures/sections/equations to have more clear separation between the three sections
CLOSED: [2025-04-08 Tue 10:46]

Prefixes:
- =detail_control_sensor=
- =detail_control_decoupling=
- =detail_control_cf=

** TODO [#A] Finish writing "multiple sensor" control section
SCHEDULED: <2025-04-08 Tue>

** TODO [#B] Review of control for Stewart platforms?

[[file:~/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/bibliography.org::*Control][Control]]

Or html version: https://research.tdehaeze.xyz/stewart-simscape/docs/bibliography.html

** TODO [#C] Discuss different strategies?

- Robust control
- Adaptive control
- etc...

* Introduction                                                        :ignore:

When controlling a MIMO system (specifically parallel manipulator such as the Stewart platform?)

- [ ] *Should the quick review of Stewart platform control be here?*
  In that case it should be possible to highlight three areas:
  - use of multiple sensors
  - decoupling strategy
  - control optimization

Several considerations:
- Section ref:sec:detail_control_sensor: How to most effectively use/combine multiple sensors
- Section ref:sec:detail_control_decoupling: How to decouple a system
- Section ref:sec:detail_control_cf: How to design the controller

* Multiple Sensor Control
:PROPERTIES:
:HEADER-ARGS:matlab+: :tangle matlab/detail_control_1_complementary_filtering.m
:END:
<<sec:detail_control_sensor>>

# file:~/Cloud/research/papers/published/dehaeze19_compl_filter_shapin_using_synth/index.org
# [[file:~/Cloud/research/papers/dehaeze21_desig_compl_filte/journal/dehaeze21_desig_compl_filte.org][file:~/Cloud/research/papers/dehaeze21_desig_compl_filte/journal/dehaeze21_desig_compl_filte.org]]
# [[file:~/Cloud/research/papers/dehaeze20_optim_robus_compl_filte/matlab/dehaeze22_optim_robus_compl_filte_matlab.org]]

** Introduction                                                      :ignore:

As was shown during the literature review of Stewart platforms, there is a large diversity of designs and included sensors and actuators.
Depending on the control objectives, which may include active damping, vibration isolation, or precise positioning, different sensor configurations are implemented.
The specific selection of the sensors, whether inertial sensors, force sensors, or relative position sensors, is heavily influenced by the control requirements of the system.

In cases where multiple control objectives must be achieved simultaneously, as is the case for the Nano Active Stabilization System (NASS) where the Stewart platform must both position the sample and provide isolation from micro-station vibrations, combining multiple sensors within the control architecture has been demonstrated to yield significant performance benefits.
From the literature, three principal approaches for combining sensors have been identified: High Authority Control-Low Authority Control (HAC-LAC), sensor fusion, and two-sensor control architectures.

#+begin_src latex :file detail_control_sensor_arch_hac_lac.pdf
\begin{tikzpicture}
  % Blocs
  \node[block={2.0cm}{2.0cm}] (P) {Plant};
  \coordinate[] (input)   at ($(P.south west)!0.5!(P.north west)$);
  \coordinate[] (outputH) at ($(P.south east)!0.2!(P.north east)$);
  \coordinate[] (outputL) at ($(P.south east)!0.8!(P.north east)$);

  \node[block, above=0.2 of P] (Klac) {$K_\text{LAC}$};
  \node[addb, left=0.5 of input] (addF) {};
  \node[block, left=0.6 of addF] (Khac) {$K_\text{HAC}$};
  % \node[addb={+}{}{}{}{-}, left=0.5 of Khac] (subr) {};

  % Connections and labels
  \draw[->] (outputL) -- ++(0.5, 0) coordinate(eastlac) |- (Klac.east);
  \node[above right] at (outputL){$x^\prime$};
  \draw[->] (Klac.west) -| (addF.north);
  \draw[->] (addF.east) -- (input) node[above left]{$u$};

  % \draw[<-] (subr.west)node[above left]{$r$} -- ++(-0.5, 0);
  % \draw[->] (outputH) -- ++(0.5, 0)  -- ++(0, -1.0) -| (subr.south);
  \draw[->] (outputH) -- ++(0.5, 0)  -- ++(0, -0.7) -| ($(Khac.west)+(-0.5, 0)$) -- (Khac.west);
  \node[above right] at (outputH){$x$};
  % \draw[->] (subr.east) -- (Khac.west) node[above left]{$\epsilon$};
  \draw[->] (Khac.east) node[above right]{$u^\prime$} -- (addF.west);

  \begin{scope}[on background layer]
    \node[fit={(Klac.north-|eastlac) (addF.west|-P.south)}, fill=black!20!white, draw, dashed, inner sep=4pt] (Pi) {};
    % \node[anchor={north west}, align=left] at (Pi.north west){\scriptsize{Damped}\\\scriptsize{Plant}};
    \node[above=0 of Pi]{\scriptsize{Damped Plant}};
  \end{scope}
\end{tikzpicture}
#+end_src

#+begin_src latex :file detail_control_sensor_arch_sensor_fusion.pdf
\begin{tikzpicture}
  % Blocs
  \node[block={2.0cm}{2.0cm}] (P) {Plant};
  \coordinate[] (input)    at ($(P.south west)!0.5!(P.north west)$);
  \coordinate[] (outputH)  at ($(P.south east)!0.2!(P.north east)$);
  \coordinate[] (outputL)  at ($(P.south east)!0.8!(P.north east)$);
  \coordinate[] (outputSS) at ($(P.south east)!0.5!(P.north east)$);

  \node[block, right=0.8 of outputH] (KH) {$H_\text{HPF}$};
  \node[block, right=0.8 of outputL] (KL) {$H_\text{LPF}$};
  \node[addb={+}{}{}{}{}, right=2.2 of outputSS] (addss) {};
  \node[block, left=0.6 of input] (K) {$K_{ss}$};

  % Connections and labels
  \draw[->] (outputL) -- (KL.west);
  \draw[->] (outputH) -- (KH.west);
  \node[above right] at (outputL){$x_L$};
  \node[above right] at (outputH){$x_H$};
  \draw[->] (KL.east) -| (addss.north);
  \draw[->] (KH.east) -| (addss.south);
  \draw[->] (addss.east) -- ++(0.9, 0);
  \draw[->] ($(addss.east) + (0.5, 0)$)node[branch]{}node[above]{$x_{ss}$} -- ++(0, -1.6) -| ($(K.west)+(-0.5, 0)$) -- (K.west);
  \draw[->] (K.east) -- (input) node[above left]{$u$};

  \begin{scope}[on background layer]
    \node[fit={(KL.north west) (KH.south-|addss.east)}, fill=black!20!white, draw, dashed, inner sep=4pt] (Pss) {};
    \node[above=0 of Pss]{\scriptsize{Sensor Fusion}};
  \end{scope}
\end{tikzpicture}
#+end_src

#+begin_src latex :file detail_control_sensor_arch_two_sensor_control.pdf
\begin{tikzpicture}
  % Blocs
  \node[block={2.0cm}{2.0cm}] (P) {Plant};
  \coordinate[] (input)    at ($(P.south west)!0.5!(P.north west)$);
  \coordinate[] (output1)  at ($(P.south east)!0.8!(P.north east)$);
  \coordinate[] (output2)  at ($(P.south east)!0.2!(P.north east)$);

  \node[addb={+}{}{}{}{}, left=0.8 of input] (addF) {};
  \coordinate[left= 1.0 of addF] (Ks);
  \node[block] (K1) at (Ks|-output1) {$K_\text{1}$};
  \node[block] (K2) at (Ks|-output2) {$K_\text{2}$};

  \draw[->] (output1) -| ++(0.6, 1.2) -| ($(K1.west)+(-0.5, 0)$) -- (K1.west);
  \draw[->] (output2) -| ++(0.6,-1.0) -| ($(K2.west)+(-0.5, 0)$) -- (K2.west);

  \draw[->] (K1.east)node[above right]{$u_1$} -| (addF.north);
  \draw[->] (K2.east)node[above right]{$u_2$} -| (addF.south);
  \draw[->] (addF.east) -- (input) node[above left=0 and 0.2]{$u$};

  \node[above right] at (output1){$x_1$};
  \node[above right] at (output2){$x_2$};

  \begin{scope}[on background layer]
    \node[fit={(K1.north west) (K2.south-|addF.east)}, fill=black!20!white, draw, dashed, inner sep=4pt] (Pss) {};
    \node[above=0 of Pss]{\scriptsize{Two-Sensor Control}};
  \end{scope}
\end{tikzpicture}
#+end_src

#+name: fig:detail_control_control_multiple_sensors
#+caption: Different control strategies when using multiple sensors. High Authority Control / Low Authority Control (\subref{fig:detail_control_sensor_arch_hac_lac}). Sensor Fusion (\subref{fig:detail_control_sensor_arch_sensor_fusion}). Two-Sensor Control (\subref{fig:detail_control_sensor_arch_two_sensor_control})
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_arch_hac_lac}HAC LAC}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
#+attr_latex: :scale 1
[[file:figs/detail_control_sensor_arch_hac_lac.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_arch_two_sensor_control}Two Sensor Control}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
#+attr_latex: :scale 1
[[file:figs/detail_control_sensor_arch_two_sensor_control.png]]
#+end_subfigure

\bigskip
#+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_arch_sensor_fusion}Sensor Fusion}
#+attr_latex: :options {0.95\textwidth}
#+begin_subfigure
#+attr_latex: :scale 1
[[file:figs/detail_control_sensor_arch_sensor_fusion.png]]
#+end_subfigure
#+end_figure

The HAC-LAC approach, used during the conceptual phase, represents a dual-loop control strategy where two control loops utilize different sensors for different purposes (Figure ref:fig:detail_control_sensor_arch_hac_lac).
In [[cite:&li01_simul_vibrat_isolat_point_contr]], vibration isolation is provided by accelerometers collocated with the voice coil actuators, while external rotational sensors are utilized to achieve pointing control.
In [[cite:&geng95_intel_contr_system_multip_degree]], force sensors collocated with the magnetostrictive actuators are used for active damping using decentralized IFF, and subsequently accelerometers are employed for adaptive vibration isolation.
Similarly, in [[cite:&wang16_inves_activ_vibrat_isolat_stewar]], piezoelectric actuators with collocated force sensors are used in a decentralized manner to provide active damping while accelerometers are implemented in an adaptive feedback loop to suppress periodic vibrations.
In [[cite:&xie17_model_contr_hybrid_passiv_activ]], force sensors are integrated in the struts for decentralized force feedback while accelerometers fixed to the top platform are employed for centralized control.

Sensor fusion, the second approach (illustrated in Figure ref:fig:detail_control_sensor_arch_sensor_fusion), involves filtering signals from two sensors using complementary filters[fn:detail_control_1] that are subsequently summed to obtain an improved sensor signal.
In [[cite:&hauge04_sensor_contr_space_based_six]], geophones (used at low frequency) are merged with force sensors (used at high frequency).
It is demonstrated that combining both sensors using sensor fusion can improve performance compared to using the individual sensors independently.
In [[cite:&tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip]], sensor fusion architecture is implemented with an accelerometer and a force sensor.
This implementation is shown to simultaneously achieve high damping of structural modes (through the force sensors) while maintaining very low vibration transmissibility (through the accelerometers).

In [[cite:&beijen14_two_sensor_contr_activ_vibrat]], the performance of sensor fusion is compared with the more general case of "two-sensor control" (illustrated in Figure ref:fig:detail_control_sensor_arch_two_sensor_control).
It is highlighted that "two-sensor control" provides greater control freedom, potentially enhancing performance.
In [[cite:&thayer02_six_axis_vibrat_isolat_system]], the use of force sensors and geophones is compared for vibration isolation purposes.
Geophones are shown to provide better isolation performance than load cells but suffer from poor robustness.
Conversely, the controller based on force sensors exhibited inferior performance (due to the presence of a pair of low frequency zeros), but demonstrated better robustness properties.
A "two-sensor control" approach was proven to perform better than controllers based on individual sensors while maintaining better robustness.
A Linear Quadratic Regulator (LQG) was employed to optimize the two-input/one-output controller.

Beyond these three main approaches, other control architectures have been proposed for different purposes.
In [[cite:&yang19_dynam_model_decoup_contr_flexib]], a first control loop utilizes force sensors and relative motion sensors to compensate for parasitic stiffness of the flexible joints.
Subsequently, the system is decoupled in the modal space (facilitated by the removal of parasitic stiffness) and accelerometers are employed for vibration isolation.

The HAC-LAC architecture was previously investigated during the conceptual phase and successfully implemented to validate the NASS concept, demonstrating excellent performance.
At the other end of the spectrum, the two-sensor approach yields greater control design freedom but introduces increased complexity in tuning, and thus was not pursued in this study.
This work instead focuses on sensor fusion, which represents a promising middle ground between the proven HAC-LAC approach and the more complex two-sensor control strategy.

A review of sensor fusion is first presented (Section ref:ssec:detail_control_sensor_review), followed by an examination of the fundamental theoretical concepts (Section ref:ssec:detail_control_sensor_fusion_requirements).
In this section, both the robustness of the fusion and the noise characteristics of the resulting "super sensor" are derived and expressed as functions of the complementary filters' norms.
A synthesis method for designing complementary filters that allow to shape their norms is proposed (Section ref:ssec:detail_control_sensor_hinf_method).
The investigation is then extended beyond the conventional two-sensor scenario, demonstrating how the proposed complementary filter synthesis can be generalized for applications requiring the fusion of three or more sensors (Section ref:ssec:detail_control_sensor_hinf_three_comp_filters).

** Review of Sensor Fusion
<<ssec:detail_control_sensor_review>>

Measuring a physical quantity using sensors is always subject to several limitations.
First, the accuracy of the measurement is affected by various noise sources, such as electrical noise from the conditioning electronics.
Second, the frequency range in which the measurement is relevant is bounded by the bandwidth of the sensor.
One way to overcome these limitations is to combine several sensors using a technique called "sensor fusion" [[cite:&bendat57_optim_filter_indep_measur_two]].
Fortunately, a wide variety of sensors exists, each with different characteristics.
By carefully selecting the sensors to be fused, a "super sensor" is obtained that combines the benefits of the individual sensors.

In some applications, sensor fusion is employed to increase measurement bandwidth [[cite:&shaw90_bandw_enhan_posit_measur_using_measur_accel;&zimmermann92_high_bandw_orien_measur_contr;&min15_compl_filter_desig_angle_estim]].
For instance, in [[cite:&shaw90_bandw_enhan_posit_measur_using_measur_accel]], the bandwidth of a position sensor is extended by fusing it with an accelerometer that provides high-frequency motion information.
In other applications, sensor fusion is utilized to obtain an estimate of the measured quantity with reduced noise [[cite:&hua05_low_ligo;&hua04_polyp_fir_compl_filter_contr_system;&plummer06_optim_compl_filter_their_applic_motion_measur;&robert12_introd_random_signal_applied_kalman]].
More recently, the fusion of sensors measuring different physical quantities has been proposed to enhance control properties [[cite:&collette15_sensor_fusion_method_high_perfor;&yong16_high_speed_vertic_posit_stage]].
In [[cite:&collette15_sensor_fusion_method_high_perfor]], an inertial sensor used for active vibration isolation is fused with a sensor collocated with the actuator to improve the stability margins of the feedback controller.

On top of Stewart platforms, practical applications of sensor fusion are numerous.
It is widely implemented for attitude estimation in autonomous vehicles such as unmanned aerial vehicles [[cite:&baerveldt97_low_cost_low_weigh_attit;&corke04_inert_visual_sensin_system_small_auton_helic;&jensen13_basic_uas]] and underwater vehicles [[cite:&pascoal99_navig_system_desig_using_time;&batista10_optim_posit_veloc_navig_filter_auton_vehic]].
Sensor fusion offers significant benefits for high-performance positioning control as demonstrated in [[cite:&shaw90_bandw_enhan_posit_measur_using_measur_accel;&zimmermann92_high_bandw_orien_measur_contr;&min15_compl_filter_desig_angle_estim;&yong16_high_speed_vertic_posit_stage]].
It has also been identified as a key technology for improving the performance of active vibration isolation systems [[cite:&tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip]].
Emblematic examples include the isolation stages of gravitational wave detectors [[cite:&collette15_sensor_fusion_method_high_perfor;&heijningen18_low]] such as those employed at LIGO [[cite:&hua05_low_ligo;&hua04_polyp_fir_compl_filter_contr_system]] and Virgo [[cite:&lucia18_low_frequen_optim_perfor_advan]].

Two principal methods are employed to perform sensor fusion: using complementary filters [[cite:&anderson53_instr_approac_system_steer_comput]] or using Kalman filtering [[cite:&brown72_integ_navig_system_kalman_filter]].
For sensor fusion applications, these methods share many relationships [[cite:&brown72_integ_navig_system_kalman_filter;&higgins75_compar_compl_kalman_filter;&robert12_introd_random_signal_applied_kalman;&fonseca15_compl]].
However, Kalman filtering requires assumptions about the probabilistic characteristics of sensor noise [[cite:&robert12_introd_random_signal_applied_kalman]], whereas complementary filters do not impose such requirements.
Furthermore, complementary filters offer advantages over Kalman filtering for sensor fusion through their general applicability, low computational cost [[cite:&higgins75_compar_compl_kalman_filter]], and intuitive nature, as their effects can be readily interpreted in the frequency domain.

A set of filters is considered complementary if the sum of their transfer functions equals one at all frequencies.
In early implementations of complementary filtering, analog circuits were used to physically realize the filters [[cite:&anderson53_instr_approac_system_steer_comput]].
While analog complementary filters remain in use today [[cite:&yong16_high_speed_vertic_posit_stage;&moore19_capac_instr_sensor_fusion_high_bandw_nanop]], digital implementation is now more common as it provides greater flexibility.

Various design methods have been developed to optimize complementary filters.
The most straightforward approach utilizes analytical formulas.
Depending on the application, these formulas may be of first order [[cite:&corke04_inert_visual_sensin_system_small_auton_helic;&yeh05_model_contr_hydraul_actuat_two;&yong16_high_speed_vertic_posit_stage]], second order [[cite:&baerveldt97_low_cost_low_weigh_attit;&stoten01_fusion_kinet_data_using_compos_filter;&jensen13_basic_uas]], or higher orders [[cite:&shaw90_bandw_enhan_posit_measur_using_measur_accel;&zimmermann92_high_bandw_orien_measur_contr;&stoten01_fusion_kinet_data_using_compos_filter;&collette15_sensor_fusion_method_high_perfor;&matichard15_seism_isolat_advan_ligo]].

Since the characteristics of the super sensor depend on the proper design of complementary filters [[cite:&dehaeze19_compl_filter_shapin_using_synth]], several optimization techniques have been developed.
Some approaches focus on finding optimal parameters for analytical formulas [[cite:&jensen13_basic_uas;&min15_compl_filter_desig_angle_estim;&fonseca15_compl]], while others employ convex optimization tools [[cite:&hua04_polyp_fir_compl_filter_contr_system;&hua05_low_ligo]] such as linear matrix inequalities [[cite:&pascoal99_navig_system_desig_using_time]].
As demonstrated in [[cite:&plummer06_optim_compl_filter_their_applic_motion_measur]], complementary filter design can be linked to the standard mixed-sensitivity control problem.
Consequently, the powerful tools developed for classical control theory can be applied to complementary filter design.
For example, in [[cite:&jensen13_basic_uas]], the two gains of a Proportional Integral (PI) controller are optimized to minimize super sensor noise.

All these complementary filter design methods share the common objective of creating a super sensor with desired characteristics, typically in terms of noise and dynamics.
As reported in [[cite:&zimmermann92_high_bandw_orien_measur_contr;&plummer06_optim_compl_filter_their_applic_motion_measur]], phase shifts and magnitude bumps in the super sensor dynamics may occur if complementary filters are poorly designed or if sensors are improperly calibrated.
Therefore, the robustness of the fusion must be considered when designing complementary filters.
Despite the numerous design methods proposed in the literature, a simple approach that specifies desired super sensor characteristics while ensuring good fusion robustness has been lacking.

Fortunately, both fusion robustness and super sensor characteristics can be linked to complementary filter magnitude [[cite:&dehaeze19_compl_filter_shapin_using_synth]].
Based on this relationship, the present work introduces an approach to designing complementary filters using $\mathcal{H}_\infty$ synthesis, which enables intuitive shaping of complementary filter magnitude in a straightforward manner.

** 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

#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src

#+begin_src matlab :tangle no :noweb yes
<<m-init-path>>
#+end_src

#+begin_src matlab :eval no :noweb yes
<<m-init-path-tangle>>
#+end_src

#+begin_src matlab :noweb yes
<<m-init-other>>

%% Initialize Frequency Vector
freqs = logspace(-1, 3, 1000);
#+end_src

** Sensor Fusion and Complementary Filters Requirements
<<ssec:detail_control_sensor_fusion_requirements>>
**** Sensor Fusion Architecture                                    :ignore:

A general sensor fusion architecture using complementary filters is shown in Figure ref:fig:detail_control_sensor_fusion_overview, where multiple sensors (in this case two) measure the same physical quantity $x$.
The sensor output signals $\hat{x}_1$ and $\hat{x}_2$ represent estimates of $x$.
These estimates are filtered by complementary filters and combined to form a new estimate $\hat{x}$.

#+begin_src latex :file detail_control_sensor_fusion_overview.pdf
\tikzset{block/.default={0.8cm}{0.8cm}}
\tikzset{addb/.append style={scale=0.7}}
\tikzset{node distance=0.6}
\begin{tikzpicture}
  \node[branch] (x) at (0, 0);
  \node[block, above right=0.3 and 0.5 of x](sensor1){Sensor 1};
  \node[block, below right=0.3 and 0.5 of x](sensor2){Sensor 2};

  \node[block, right=1.1 of sensor1](H1){$H_1(s)$};
  \node[block, right=1.1 of sensor2](H2){$H_2(s)$};

  \node[addb, right=5.0 of x](add){};

  \draw[] ($(x)+(-0.7, 0)$) node[above right]{$x$} -- (x.center);
  \draw[->] (x.center) |- (sensor1.west);
  \draw[->] (x.center) |- (sensor2.west);
  \draw[->] (sensor1.east) -- node[midway, above]{$\hat{x}_1$} (H1.west);
  \draw[->] (sensor2.east) -- node[midway, above]{$\hat{x}_2$} (H2.west);
  \draw[->] (H1) -| (add.north);
  \draw[->] (H2) -| (add.south);
  \draw[->] (add.east) -- ++(0.9, 0) node[above left]{$\hat{x}$};

  \begin{scope}[on background layer]
    \node[fit={($(H2.south-|x) + (0, -0.2)$) ($(H1.north-|add.east) + (0.2, 0.6)$)}, fill=black!10!white, draw, inner sep=6pt] (supersensor) {};
    \node[below] at (supersensor.north) {Super Sensor};

    \node[fit={(sensor2.south west) (sensor1.north east)}, fill=black!20!white, draw, inner sep=6pt] (sensors) {};
    \node[align=center] at (sensors.center) {{\tiny Normalized}\\[-0.5em]{\tiny Sensors}};

    \node[fit={(H2.south west) (H1.north-|add.east)}, fill=black!20!white, draw, inner sep=6pt] (filters) {};
    \node[align=center] at ($(filters.center) + (-0.3, 0)$) {{\tiny Complementary}\\[-0.5em]{\tiny Filters}};
  \end{scope}
\end{tikzpicture}
#+end_src

#+name: fig:detail_control_sensor_fusion_overview
#+caption: Schematic of a sensor fusion architecture using complementary filters.
#+RESULTS:
[[file:figs/detail_control_sensor_fusion_overview.png]]

The complementary property of filters $H_1(s)$ and $H_2(s)$ requires that the sum of their transfer functions equals one at all frequencies eqref:eq:detail_control_sensor_comp_filter.

\begin{equation}\label{eq:detail_control_sensor_comp_filter}
  H_1(s) + H_2(s) = 1
\end{equation}

**** Sensor Models and Sensor Normalization

To analyze sensor fusion architectures, appropriate sensor models are required.
The model shown in Figure ref:fig:detail_control_sensor_model consists of a linear time invariant (LTI) system $G_i(s)$ representing the sensor dynamics and an input $n_i$ representing sensor noise.
The model input $x$ is the measured physical quantity, and its output $\tilde{x}_i$ is the "raw" output of the sensor.

Prior to filtering the sensor outputs $\tilde{x}_i$ with complementary filters, the sensors are typically normalized to simplify the fusion process.
This normalization involves using an estimate $\hat{G}_i(s)$ of the sensor dynamics $G_i(s)$, and filtering the sensor output by the inverse of this estimate $\hat{G}_i^{-1}(s)$, as shown in Figure ref:fig:detail_control_sensor_model_calibrated.
It is assumed that the sensor inverse $\hat{G}_i^{-1}(s)$ is proper and stable.
This approach ensures that the units of the estimates $\hat{x}_i$ match the units of the physical quantity $x$.
The sensor dynamics estimate $\hat{G}_i(s)$ may be a simple gain or a more complex transfer function.

#+begin_src latex :file detail_control_sensor_model.pdf
\tikzset{block/.default={0.8cm}{0.8cm}}
\tikzset{addb/.append style={scale=0.7}}
\tikzset{node distance=0.6}
\begin{tikzpicture}
  \node[addb](add1){};
  \node[block, right=0.8 of add1](G1){$G_i(s)$};

  \draw[->] ($(add1.west)+(-0.7, 0)$) node[above right]{$x$} -- (add1.west);
  \draw[<-] (add1.north) -- ++(0, 0.7)node[below right](n1){$n_i$};
  \draw[->] (add1.east) -- (G1.west);
  \draw[->] (G1.east) -- ++(0.7, 0) node[above left]{$\tilde{x}_i$};

  \begin{scope}[on background layer]
    \node[fit={(add1.west |- G1.south) (n1.north -| G1.east)}, fill=black!20!white, draw, inner sep=3pt] (sensor1) {};
    \node[below left] at (sensor1.north east) {Sensor};
  \end{scope}
\end{tikzpicture}
#+end_src

#+begin_src latex :file detail_control_sensor_model_calibrated.pdf
\tikzset{block/.default={0.8cm}{0.8cm}}
\tikzset{addb/.append style={scale=0.7}}
\tikzset{node distance=0.6}
\begin{tikzpicture}
  \node[addb](add1){};
  \node[block, right=0.8 of add1](G1){$G_i(s)$};
  \node[block, right=0.8 of G1](G1inv){$\hat{G}_i^{-1}(s)$};

  \draw[->] ($(add1.west)+(-0.7, 0)$) node[above right]{$x$} -- (add1.west);
  \draw[<-] (add1.north) -- ++(0, 0.7)node[below right](n1){$n_i$};
  \draw[->] (add1.east) -- (G1.west);
  \draw[->] (G1.east) -- (G1inv.west) node[above left]{$\tilde{x}_i$};
  \draw[->] (G1inv.east) -- ++(0.8, 0) node[above left]{$\hat{x}_i$};

  \begin{scope}[on background layer]
    \node[fit={(add1.west |- G1inv.south) (n1.north -| G1inv.east)}, fill=black!10!white, draw, inner sep=6pt] (sensor1cal) {};
    \node[below left, align=right] at (sensor1cal.north east) {{\tiny Normalized}\\[-0.5em]{\tiny sensor}};

    \node[fit={(add1.west |- G1.south) (n1.north -| G1.east)}, fill=black!20!white, draw, inner sep=3pt] (sensor1) {};
    \node[below left] at (sensor1.north east) {Sensor};
  \end{scope}
\end{tikzpicture}
#+end_src

#+name: fig:detail_control_sensor_models
#+caption: Sensor models with and without normalization.
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_model}Basic sensor model consisting of a noise input $n_i$ and a linear time invariant transfer function $G_i(s)$}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
#+attr_latex: :scale 1
[[file:figs/detail_control_sensor_model.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_model_calibrated}Normalized sensors using the inverse of an estimate $\hat{G}}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
#+attr_latex: :scale 1
[[file:figs/detail_control_sensor_model_calibrated.png]]
#+end_subfigure
#+end_figure

Two normalized sensors are then combined to form a super sensor as shown in Figure ref:fig:detail_control_sensor_fusion_super_sensor.
The two sensors measure the same physical quantity $x$ with dynamics $G_1(s)$ and $G_2(s)$, and with uncorrelated noises $n_1$ and $n_2$.
The signals from both normalized sensors are fed into two complementary filters $H_1(s)$ and $H_2(s)$ and then combined to yield an estimate $\hat{x}$ of $x$.
The super sensor output $\hat{x}$ is therefore described by eqref:eq:detail_control_sensor_comp_filter_estimate.

\begin{equation}\label{eq:detail_control_sensor_comp_filter_estimate}
  \hat{x} = \Big( H_1(s) \hat{G}_1^{-1}(s) G_1(s) + H_2(s) \hat{G}_2^{-1}(s) G_2(s) \Big) x + H_1(s) \hat{G}_1^{-1}(s) G_1(s) n_1 + H_2(s) \hat{G}_2^{-1}(s) G_2(s) n_2
\end{equation}

#+begin_src latex :file detail_control_sensor_fusion_super_sensor.pdf
\tikzset{block/.default={0.8cm}{0.8cm}}
\tikzset{addb/.append style={scale=0.7}}
\tikzset{node distance=0.6}
\begin{tikzpicture}
  \node[branch] (x) at (0, 0);
  \node[addb, above right=0.8 and 0.5 of x](add1){};
  \node[addb, below right=0.8 and 0.5 of x](add2){};
  \node[block, right=0.8 of add1](G1){$G_1(s)$};
  \node[block, right=0.8 of add2](G2){$G_2(s)$};
  \node[block, right=0.8 of G1](G1inv){$\hat{G}_1^{-1}(s)$};
  \node[block, right=0.8 of G2](G2inv){$\hat{G}_2^{-2}(s)$};
  \node[block, right=0.8 of G1inv](H1){$H_1(s)$};
  \node[block, right=0.8 of G2inv](H2){$H_2(s)$};
  \node[addb, right=7 of x](add){};

  \draw[] ($(x)+(-0.7, 0)$) node[above right]{$x$} -- (x.center);
  \draw[->] (x.center) |- (add1.west);
  \draw[->] (x.center) |- (add2.west);
  \draw[<-] (add1.north) -- ++(0, 0.7)node[below right](n1){$n_1$};
  \draw[->] (add1.east) -- (G1.west);
  \draw[->] (G1.east) -- (G1inv.west) node[above left]{$\tilde{x}_1$};
  \draw[->] (G1inv.east) -- (H1.west) node[above left]{$\hat{x}_1$};
  \draw[<-] (add2.north) -- ++(0, 0.7)node[below right](n2){$n_2$};
  \draw[->] (add2.east) -- (G2.west);
  \draw[->] (G2.east) -- (G2inv.west) node[above left]{$\tilde{x}_2$};
  \draw[->] (G2inv.east) -- (H2.west) node[above left]{$\hat{x}_2$};
  \draw[->] (H1) -| (add.north);
  \draw[->] (H2) -| (add.south);
  \draw[->] (add.east) -- ++(0.7, 0) node[above left]{$\hat{x}$};

  \begin{scope}[on background layer]
    \node[fit={(G2.south-|x) (n1.north-|add.east)}, fill=black!10!white, draw, inner sep=9pt] (supersensor) {};
    \node[below left] at (supersensor.north east) {Super Sensor};

    \node[fit={(add1.west |- G1inv.south) (n1.north -| G1inv.east)}, fill=colorblue!20!white, draw, inner sep=6pt] (sensor1cal) {};
    \node[below left, align=right] at (sensor1cal.north east) {{\tiny Normalized}\\[-0.5em]{\tiny sensor}};
    \node[fit={(add1.west |- G1.south) (n1.north -| G1.east)}, fill=colorblue!30!white, draw, inner sep=3pt] (sensor1) {};
    \node[below left] at (sensor1.north east) {Sensor 1};

    \node[fit={(add2.west |- G2inv.south) (n2.north -| G2inv.east)}, fill=colorred!20!white, draw, inner sep=6pt] (sensor2cal) {};
    \node[below left, align=right] at (sensor2cal.north east) {{\tiny Normalized}\\[-0.5em]{\tiny sensor}};
    \node[fit={(add2.west |- G2.south) (n2.north -| G2.east)}, fill=colorred!30!white, draw, inner sep=3pt] (sensor2) {};
    \node[below left] at (sensor2.north east) {Sensor 2};
  \end{scope}
\end{tikzpicture}
#+end_src

#+name: fig:detail_control_sensor_fusion_super_sensor
#+caption: Sensor fusion architecture with two normalized sensors.
#+RESULTS:
[[file:figs/detail_control_sensor_fusion_super_sensor.png]]

**** Noise Sensor Filtering

First, consider the case where all sensors are perfectly normalized eqref:eq:detail_control_sensor_perfect_dynamics.
The effects of imperfect normalization will be addressed subsequently.

\begin{equation}\label{eq:detail_control_sensor_perfect_dynamics}
  \frac{\hat{x}_i}{x} = \hat{G}_i(s) G_i(s) = 1
\end{equation}

In that case, the super sensor output $\hat{x}$ equals $x$ plus the filtered noise from both sensors eqref:eq:detail_control_sensor_estimate_perfect_dyn.
From this equation, it is evident that the complementary filters $H_1(s)$ and $H_2(s)$ operate solely on the sensor noise.
Thus, this sensor fusion architecture allows filtering of sensor noise without introducing distortion in the measured physical quantity.
This fundamental property necessitates that the two filters must be complementary.

\begin{equation}\label{eq:detail_control_sensor_estimate_perfect_dyn}
  \hat{x} = x + H_1(s) n_1 + H_2(s) n_2
\end{equation}

The estimation error $\epsilon_x$, defined as the difference between the sensor output $\hat{x}$ and the measured quantity $x$, is computed for the super sensor eqref:eq:detail_control_sensor_estimate_error.

\begin{equation}\label{eq:detail_control_sensor_estimate_error}
  \epsilon_x \triangleq \hat{x} - x = H_1(s) n_1 + H_2(s) n_2
\end{equation}

As shown in eqref:eq:detail_control_sensor_noise_filtering_psd, the Power Spectral Density (PSD) of the estimation error $\Phi_{\epsilon_x}$ depends both on the norm of the two complementary filters and on the PSD of the noise sources $\Phi_{n_1}$ and $\Phi_{n_2}$.

\begin{equation}\label{eq:detail_control_sensor_noise_filtering_psd}
  \Phi_{\epsilon_x}(\omega) = \left|H_1(j\omega)\right|^2 \Phi_{n_1}(\omega) + \left|H_2(j\omega)\right|^2 \Phi_{n_2}(\omega)
\end{equation}

If the two sensors have identical noise characteristics ($\Phi_{n_1}(\omega) = \Phi_{n_2}(\omega)$), simple averaging ($H_1(s) = H_2(s) = 0.5$) would minimize the super sensor noise.
This represents the simplest form of sensor fusion using complementary filters.

However, sensors typically exhibit high noise levels in different frequency regions.
In such cases, to reduce the noise of the super sensor, the norm $|H_1(j\omega)|$ should be minimized when $\Phi_{n_1}(\omega)$ exceeds $\Phi_{n_2}(\omega)$, and the norm $|H_2(j\omega)|$ should be minimized when $\Phi_{n_2}(\omega)$ exceeds $\Phi_{n_1}(\omega)$.
Therefore, by appropriately shaping the norm of the complementary filters, the noise of the super sensor can be minimized.

**** Sensor Fusion Robustness

In practical systems, sensor normalization is rarely perfect, and condition eqref:eq:detail_control_sensor_perfect_dynamics is not fully satisfied.

To analyze such imperfections, a multiplicative input uncertainty is incorporated into the sensor dynamics (Figure ref:fig:detail_control_sensor_model_uncertainty).
The nominal model is the estimated model used for normalization $\hat{G}_i(s)$, $\Delta_i(s)$ is any stable transfer function satisfying $|\Delta_i(j\omega)| \le 1,\ \forall\omega$, and $w_i(s)$ is a weighting transfer function representing the magnitude of uncertainty.
The weight $w_i(s)$ is selected such that the actual sensor dynamics $G_i(j\omega)$ remains within the uncertain region represented by a circle in the complex plane, centered on $1$ with a radius equal to $|w_i(j\omega)|$.

Since the nominal sensor dynamics is taken as the normalized filter, the normalized sensor model can be further simplified as shown in Figure ref:fig:detail_control_sensor_model_uncertainty_simplified.

#+begin_src latex :file detail_control_sensor_model_uncertainty.pdf
\tikzset{block/.default={0.8cm}{0.8cm}}
\tikzset{addb/.append style={scale=0.7}}
\tikzset{node distance=0.6}
\begin{tikzpicture}
  \node[branch] (input) at (0,0) {};
  \node[block, above right= 0.4 and 0.4 of input](W1){$w_1(s)$};
  \node[block, right=0.4 of W1](delta1){$\Delta_1(s)$};
  \node[addb] (addu) at ($(delta1.east|-input) + (0.4, 0)$) {};
  \node[addb, right=0.4 of addu] (addn) {};
  \node[block, right=0.4 of addn] (G1) {$\hat{G}_1(s)$};
  \node[block, right=0.8 of G1](G1inv){$\hat{G}_1^{-1}(s)$};

  \draw[->] ($(input)+(-0.7, 0)$) node[above right]{$x$} -- (addu);
  \draw[->] (input.center) |- (W1.west);
  \draw[->] (W1.east) -- (delta1.west);
  \draw[->] (delta1.east) -| (addu.north);
  \draw[->] (addu.east) -- (addn.west);
  \draw[->] (addn.east) -- (G1.west);
  \draw[<-] (addn.north) -- ++(0, 0.7)node[below right](n1){$n_1$};
  \draw[->] (G1.east) -- (G1inv.west) node[above left]{$\tilde{x}_1$};
  \draw[->] (G1inv.east) -- ++(0.8, 0) node[above left]{$\hat{x}_1$};

  \begin{scope}[on background layer]
    \node[fit={(input.west |- G1inv.south) (delta1.north -| G1inv.east)}, fill=black!10!white, draw, inner sep=6pt] (sensor1cal) {};
    \node[below left, align=right] at (sensor1cal.north east) {{\tiny Normalized}\\[-0.5em]{\tiny sensor}};

    \node[fit={(input.west |- G1.south) (delta1.north -| G1.east)}, fill=black!20!white, draw, inner sep=3pt] (sensor1) {};
    \node[below left] at (sensor1.north east) {Sensor};
  \end{scope}
\end{tikzpicture}
#+end_src

#+begin_src latex :file detail_control_sensor_model_uncertainty_simplified.pdf
\tikzset{block/.default={0.8cm}{0.8cm}}
\tikzset{addb/.append style={scale=0.7}}
\tikzset{node distance=0.6}
\begin{tikzpicture}
  \node[branch] (input) at (0,0) {};
  \node[block, above right= 0.4 and 0.4 of input](W1){$w_1(s)$};
  \node[block, right=0.4 of W1](delta1){$\Delta_1(s)$};
  \node[addb] (addu) at ($(delta1.east|-input) + (0.4, 0)$) {};
  \node[addb, right=0.4 of addu] (addn) {};

  \draw[->] ($(input)+(-0.8, 0)$) node[above right]{$x$} -- (addu);
  \draw[->] (input.center) |- (W1.west);
  \draw[->] (W1.east) -- (delta1.west);
  \draw[->] (delta1.east) -| (addu.north);
  \draw[->] (addu.east) -- (addn.west);
  \draw[<-] (addn.north) -- ++(0, 0.6)node[below right](n1){$n_1$};
  \draw[->] (addn.east) -- ++(0.9, 0) node[above left]{$\hat{x}_1$};

  \begin{scope}[on background layer]
    \node[fit={(input.west |- addu.south) ($(delta1.north -| addn.east) + (0.1, 0)$)}, fill=black!10!white, draw, inner sep=6pt] (sensor1cal) {};
    \node[below left, align=right] at (sensor1cal.north east) {{\tiny Normalized}\\[-0.5em]{\tiny sensor}};
  \end{scope}
\end{tikzpicture}
#+end_src

#+name: fig:detail_control_sensor_models_uncertainty
#+caption: Sensor models with dynamical uncertainty
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_model_uncertainty}Sensor with multiplicative input uncertainty}
#+attr_latex: :options {0.58\textwidth}
#+begin_subfigure
#+attr_latex: :width 0.95\linewidth
[[file:figs/detail_control_sensor_model_uncertainty.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_model_uncertainty_simplified}Simplified sensor model}
#+attr_latex: :options {0.38\textwidth}
#+begin_subfigure
#+attr_latex: :width 0.95\linewidth
[[file:figs/detail_control_sensor_model_uncertainty_simplified.png]]
#+end_subfigure
#+end_figure

The sensor fusion architecture incorporating sensor models with dynamical uncertainty is illustrated in Figure ref:fig:detail_control_sensor_fusion_dynamic_uncertainty.
The super sensor dynamics eqref:eq:detail_control_sensor_super_sensor_dyn_uncertainty is no longer unity but depends on the sensor dynamical uncertainty weights $w_i(s)$ and the complementary filters $H_i(s)$.
The dynamical uncertainty of the super sensor can be graphically represented in the complex plane by a circle centered on $1$ with a radius equal to $|w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)|$ (Figure ref:fig:detail_control_sensor_uncertainty_set_super_sensor).


The sensor fusion architecture with the sensor models including dynamical uncertainty is shown in Figure ref:fig:detail_control_sensor_fusion_dynamic_uncertainty.
The super sensor dynamics eqref:eq:detail_control_sensor_super_sensor_dyn_uncertainty is no longer equal to $1$ and now depends on the sensor dynamical uncertainty weights $w_i(s)$ as well as on the complementary filters $H_i(s)$.
The dynamical uncertainty of the super sensor can be graphically represented in the complex plane by a circle centered on $1$ with a radius equal to $|w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)|$ (Figure ref:fig:detail_control_sensor_uncertainty_set_super_sensor).

\begin{equation}\label{eq:detail_control_sensor_super_sensor_dyn_uncertainty}
  \frac{\hat{x}}{x} = 1 + w_1(s) H_1(s) \Delta_1(s) + w_2(s) H_2(s) \Delta_2(s)
\end{equation}

#+begin_src latex :file detail_control_sensor_fusion_dynamic_uncertainty.pdf
\tikzset{block/.default={0.8cm}{0.8cm}}
\tikzset{addb/.append style={scale=0.7}}
\tikzset{node distance=0.6}
\begin{tikzpicture}
  \node[branch] (x) at (0, 0);

  \node[branch, above right=1.0 and 0.3 of x] (input1) {};
  \node[branch, below right=1.0 and 0.3 of x] (input2) {};
  \node[block, above right= 0.4 and 0.3 of input1](W1){$w_1(s)$};
  \node[block, above right= 0.4 and 0.3 of input2](W2){$w_2(s)$};
  \node[block, right=0.4 of W1](delta1){$\Delta_1(s)$};
  \node[block, right=0.4 of W2](delta2){$\Delta_2(s)$};
  \node[addb] (addu1) at ($(delta1.east|-input1) + (0.4, 0)$) {};
  \node[addb] (addu2) at ($(delta2.east|-input2) + (0.4, 0)$) {};
  \node[addb, right=0.4 of addu1] (addn1) {};
  \node[addb, right=0.4 of addu2] (addn2) {};
  \node[block, right=0.9 of addn1](H1){$H_1(s)$};
  \node[block, right=0.9 of addn2](H2){$H_2(s)$};

  \node[addb, right=7 of x](add){};


  \draw[] ($(x)+(-0.7, 0)$) node[above right]{$x$} -- (x.center);
  \draw[->] (x.center) |- (addu1.west);
  \draw[->] (x.center) |- (addu2.west);
  \draw[->] (input1.center) |- (W1.west);
  \draw[->] (W1.east) -- (delta1.west);
  \draw[->] (delta1.east) -| (addu1.north);
  \draw[->] (addu1.east) -- (addn1.west);
  \draw[<-] (addn1.north) -- ++(0, 0.6)node[below right](n1){$n_1$};
  \draw[->] (input2.center) |- (W2.west);
  \draw[->] (W2.east) -- (delta2.west);
  \draw[->] (delta2.east) -| (addu2.north);
  \draw[->] (addu2.east) -- (addn2.west);
  \draw[<-] (addn2.north) -- ++(0, 0.6)node[below right](n2){$n_2$};

  \draw[->] (addn1.east) -- (H1.west) node[above left]{$\hat{x}_1$};
  \draw[->] (addn2.east) -- (H2.west) node[above left]{$\hat{x}_2$};
  \draw[->] (H1) -| (add.north);
  \draw[->] (H2) -| (add.south);
  \draw[->] (add.east) -- ++(0.7, 0) node[above left]{$\hat{x}$};

  \begin{scope}[on background layer]
    \node[fit={(addn2.south-|x) (delta1.north-|add.east)}, fill=black!10!white, draw, inner sep=9pt] (supersensor) {};
    \node[below left] at (supersensor.north east) {Super Sensor};

    \node[fit={(input1.west |- addu1.south) ($(delta1.north -| addn1.east) + (0.1, 0.0)$)}, fill=colorblue!20!white, draw, inner sep=6pt] (sensor1cal) {};
    \node[below left, align=right] at (sensor1cal.north east) {{\tiny Normalized}\\[-0.5em]{\tiny sensor 1}};

    \node[fit={(input2.west |- addu2.south) ($(delta2.north -| addn1.east) + (0.1, 0.0)$)}, fill=colorred!20!white, draw, inner sep=6pt] (sensor2cal) {};
    \node[below left, align=right] at (sensor2cal.north east) {{\tiny Normalized}\\[-0.5em]{\tiny sensor 2}};
  \end{scope}
\end{tikzpicture}
#+end_src

#+begin_src latex :file detail_control_sensor_uncertainty_set_super_sensor.pdf
\tikzset{block/.default={0.8cm}{0.8cm}}
\tikzset{addb/.append style={scale=0.7}}
\tikzset{node distance=0.6}
\begin{tikzpicture}
  \begin{scope}[shift={(4, 0)}]

    % Uncertainty Circle
    \node[draw, circle, fill=black!20!white, minimum size=3.6cm] (c) at (0, 0) {};
    \path[draw, fill=colorblue!20!white] (0, 0) circle [radius=1.0];
    \path[draw, fill=colorred!20!white] (135:1.0) circle [radius=0.8];
    \path[draw, dashed] (0, 0) circle [radius=1.0];

    % Center of Circle
    \node[below] at (0, 0){$1$};

    \draw[<->] (0, 0)   node[branch]{} -- coordinate[midway](r1) ++(45:1.0);
    \draw[<->] (135:1.0)node[branch]{} -- coordinate[midway](r2) ++(135:0.8);

    \node[] (l1) at (2, 1.5) {$|w_1 H_1|$};
    \draw[->, out=-90, in=0] (l1.south) to (r1);

    \node[] (l2) at (-3.2, 1.2) {$|w_2 H_2|$};
    \draw[->, out=0, in=-180] (l2.east) to (r2);

    \draw[<->] (0, 0) -- coordinate[near end](r3) ++(200:1.8);
    \node[] (l3) at (-2.5, -1.5) {$|w_1 H_1| + |w_2 H_2|$};
    \draw[->, out=90, in=-90] (l3.north) to (r3);
  \end{scope}

  % Real and Imaginary Axis
  \draw[->] (-0.5, 0) -- (7.0, 0) node[below left]{Re};
  \draw[->] (0, -1.7) -- (0, 1.7) node[below left]{Im};

  \draw[dashed] (0, 0) -- (tangent cs:node=c,point={(0, 0)},solution=2);
  \draw[dashed] (1, 0) arc (0:28:1) node[midway, right]{$\Delta \phi_\text{max}$};
\end{tikzpicture}
#+end_src

#+name: fig:detail_control_sensor_uncertainty
#+caption: Sensor fusion architecture with sensor dynamics uncertainty (\subref{fig:detail_control_sensor_fusion_dynamic_uncertainty}). Uncertainty region (\subref{fig:detail_control_sensor_uncertainty_set_super_sensor}) of the super sensor dynamics in the complex plane (grey circle). The contribution of both sensors 1 and 2 to the total uncertainty are represented respectively by a blue circle and a red circle. The frequency dependency $\omega$ is here omitted.
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_fusion_dynamic_uncertainty}Sensor Fusion Architecture}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
#+attr_latex: :width 0.95\linewidth
[[file:figs/detail_control_sensor_fusion_dynamic_uncertainty.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_uncertainty_set_super_sensor}Uncertainty regions}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
#+attr_latex: :width 0.95\linewidth
[[file:figs/detail_control_sensor_uncertainty_set_super_sensor.png]]
#+end_subfigure
#+end_figure

The super sensor dynamical uncertainty, and consequently the robustness of the fusion, clearly depends on the complementary filters' norm.
As it is generally desired to limit the dynamical uncertainty of the super sensor, 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.

** Complementary Filters Shaping
<<ssec:detail_control_sensor_hinf_method>>
**** Introduction                                                  :ignore:

As demonstrated in Section ref:ssec:detail_control_sensor_fusion_requirements, both the noise characteristics and robustness of the super sensor are functions of the complementary filters' norm.
Consequently, a synthesis method that enables precise shaping of complementary filter norms would provide significant practical benefits.
In this section, such a synthesis approach is developed by formulating the design objective as a standard $\mathcal{H}_\infty$ optimization problem.
The proper design of weighting functions, which are used to specify the desired complementary filter shapes during synthesis, is discussed in detail.
Finally, the efficacy of the proposed synthesis method is validated through a simple example.

**** Synthesis Objective

The primary objective is to shape the norms of two filters $H_1(s)$ and $H_2(s)$ while ensuring they maintain their complementary property as defined in eqref:eq:detail_control_sensor_comp_filter.
This is equivalent to finding proper and stable transfer functions $H_1(s)$ and $H_2(s)$ that satisfy conditions eqref:eq:detail_control_sensor_hinf_cond_complementarity, eqref:eq:detail_control_sensor_hinf_cond_h1, and eqref:eq:detail_control_sensor_hinf_cond_h2.
The functions $W_1(s)$ and $W_2(s)$ represent weighting transfer functions that are carefully selected to specify the maximum desired norm of the complementary filters during synthesis.

\begin{subequations}\label{eq:detail_control_sensor_comp_filter_problem_form}
  \begin{align}
  & H_1(s) + H_2(s) = 1 \label{eq:detail_control_sensor_hinf_cond_complementarity} \\
  & |H_1(j\omega)| \le \frac{1}{|W_1(j\omega)|} \quad \forall\omega \label{eq:detail_control_sensor_hinf_cond_h1} \\
  & |H_2(j\omega)| \le \frac{1}{|W_2(j\omega)|} \quad \forall\omega \label{eq:detail_control_sensor_hinf_cond_h2}
  \end{align}
\end{subequations}

**** Shaping of Complementary Filters using $\mathcal{H}_\infty$ synthesis

The synthesis objective can be readily expressed as a standard $\mathcal{H}_\infty$ optimization problem and solved using widely available computational tools.
Consider the generalized plant $P(s)$ illustrated in Figure ref:fig:detail_control_sensor_h_infinity_robust_fusion_plant and mathematically described by eqref:eq:detail_control_sensor_generalized_plant.

\begin{equation}\label{eq:detail_control_sensor_generalized_plant}
  \begin{bmatrix} z_1 \\ z_2 \\ v \end{bmatrix} = P(s) \begin{bmatrix} w\\u \end{bmatrix}; \quad P(s) = \begin{bmatrix}W_1(s) & -W_1(s) \\ 0 & \phantom{+}W_2(s) \\  1 & 0 \end{bmatrix}
\end{equation}

#+begin_src latex :file detail_control_sensor_h_infinity_robust_fusion_plant.pdf
\tikzset{block/.default={0.8cm}{0.8cm}}
\tikzset{addb/.append style={scale=0.7}}
\tikzset{node distance=0.6}
\begin{tikzpicture}
  \node[block={4.0cm}{3.0cm}, fill=black!10!white] (P) {};
  \node[above] at (P.north) {$P(s)$};

  \node[block, below=0.2 of P, opacity=0] (H2) {$H_2(s)$};

  \coordinate[] (inputw)  at ($(P.south west)!0.75!(P.north west) + (-0.7, 0)$);
  \coordinate[] (inputu)  at ($(P.south west)!0.35!(P.north west) + (-0.7, 0)$);

  \coordinate[] (output1) at ($(P.south east)!0.75!(P.north east) + ( 0.7, 0)$);
  \coordinate[] (output2) at ($(P.south east)!0.35!(P.north east) + ( 0.7, 0)$);
  \coordinate[] (outputv) at ($(P.south east)!0.1!(P.north east) + ( 0.7, 0)$);

  \node[block, left=1.4 of output1] (W1){$W_1(s)$};
  \node[block, left=1.4 of output2] (W2){$W_2(s)$};
  \node[addb={+}{}{}{}{-}, left=of W1] (sub) {};

  \draw[->] (inputw) node[above right]{$w$} -- (sub.west);
  \draw[->] (inputu) node[above right]{$u$} -- (W2.west);
  \draw[->] (inputu-|sub) node[branch]{} -- (sub.south);
  \draw[->] (sub.east) -- (W1.west);
  \draw[->] ($(sub.west)+(-0.6, 0)$) node[branch]{} |- (outputv) node[above left]{$v$};
  \draw[->] (W1.east) -- (output1)node[above left]{$z_1$};
  \draw[->] (W2.east) -- (output2)node[above left]{$z_2$};
\end{tikzpicture}
#+end_src

#+begin_src latex :file detail_control_sensor_h_infinity_robust_fusion_fb.pdf
\tikzset{block/.default={0.8cm}{0.8cm}}
\tikzset{addb/.append style={scale=0.7}}
\tikzset{node distance=0.6}
\begin{tikzpicture}
  \node[block={4.0cm}{3.0cm}, fill=black!10!white] (P) {};
  \node[above] at (P.north) {$P(s)$};

  \node[block, below=0.2 of P] (H2) {$H_2(s)$};

  \coordinate[] (inputw)  at ($(P.south west)!0.75!(P.north west) + (-0.7, 0)$);
  \coordinate[] (inputu)  at ($(P.south west)!0.35!(P.north west) + (-0.4, 0)$);

  \coordinate[] (output1) at ($(P.south east)!0.75!(P.north east) + ( 0.7, 0)$);
  \coordinate[] (output2) at ($(P.south east)!0.35!(P.north east) + ( 0.7, 0)$);
  \coordinate[] (outputv) at ($(P.south east)!0.1!(P.north east) + ( 0.4, 0)$);

  \node[block, left=1.4 of output1] (W1){$W_1(s)$};
  \node[block, left=1.4 of output2] (W2){$W_2(s)$};
  \node[addb={+}{}{}{}{-}, left=of W1] (sub) {};

  \draw[->] (inputw) node[above right]{$w$} -- (sub.west);
  \draw[->] (inputu-|sub) node[branch]{} -- (sub.south);
  \draw[->] (sub.east) -- (W1.west);
  \draw[->] ($(sub.west)+(-0.6, 0)$) node[branch]{} |- (outputv) |- (H2.east);
  \draw[->] (H2.west) -| (inputu) -- (W2.west);
  \draw[->] (W1.east) -- (output1)node[above left]{$z_1$};
  \draw[->] (W2.east) -- (output2)node[above left]{$z_2$};
\end{tikzpicture}
#+end_src

#+name: fig:detail_control_sensor_h_infinity_robust_fusion
#+caption: Architecture for the $\mathcal{H}_\infty$ synthesis of complementary filters
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_h_infinity_robust_fusion_plant}Generalized plant}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
#+attr_latex: :scale 1
[[file:figs/detail_control_sensor_h_infinity_robust_fusion_plant.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_h_infinity_robust_fusion_fb}Generalized plant with the synthesized filter}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
#+attr_latex: :scale 1
[[file:figs/detail_control_sensor_h_infinity_robust_fusion_fb.png]]
#+end_subfigure
#+end_figure

Applying standard $\mathcal{H}_\infty$ synthesis to the generalized plant $P(s)$ is equivalent to finding a stable filter $H_2(s)$ that, based on input $v$, generates an output signal $u$ such that the $\mathcal{H}_\infty$ norm of the system shown in Figure ref:fig:detail_control_sensor_h_infinity_robust_fusion_fb from $w$ to $[z_1, \ z_2]$ does not exceed unity, as expressed in eqref:eq:detail_control_sensor_hinf_syn_obj.

\begin{equation}\label{eq:detail_control_sensor_hinf_syn_obj}
  \left\|\begin{matrix} \left(1 - H_2(s)\right) W_1(s) \\ H_2(s) W_2(s) \end{matrix}\right\|_\infty \le 1
\end{equation}

By defining $H_1(s)$ as the complement of $H_2(s)$ (eqref:eq:detail_control_sensor_definition_H1), the $\mathcal{H}_\infty$ synthesis objective becomes equivalent to eqref:eq:detail_control_sensor_hinf_problem, ensuring that conditions eqref:eq:detail_control_sensor_hinf_cond_h1 and eqref:eq:detail_control_sensor_hinf_cond_h2 are satisfied.

\begin{equation}\label{eq:detail_control_sensor_definition_H1}
  H_1(s) \triangleq 1 - H_2(s)
\end{equation}

\begin{equation}\label{eq:detail_control_sensor_hinf_problem}
  \left\|\begin{matrix} H_1(s) W_1(s) \\ H_2(s) W_2(s) \end{matrix}\right\|_\infty \le 1
\end{equation}

Therefore, applying $\mathcal{H}_\infty$ synthesis to the standard plant $P(s)$ (eqref:eq:detail_control_sensor_generalized_plant) generates two filters, $H_2(s)$ and $H_1(s) \triangleq 1 - H_2(s)$, that are complementary as required by eqref:eq:detail_control_sensor_comp_filter_problem_form, with norms bounded by the specified constraints in eqref:eq:detail_control_sensor_hinf_cond_h1 and eqref:eq:detail_control_sensor_hinf_cond_h2.

It should be noted that there exists only an implication (not an equivalence) between the $\mathcal{H}_\infty$ norm condition in eqref:eq:detail_control_sensor_hinf_problem and the initial synthesis objectives in eqref:eq:detail_control_sensor_hinf_cond_h1 and eqref:eq:detail_control_sensor_hinf_cond_h2.
Consequently, the optimization may be somewhat conservative with respect to the set of filters on which it operates (see [[cite:&skogestad07_multiv_feedb_contr,Chap. 2.8.3]]).

**** Weighting Functions Design

Weighting functions play a crucial role during synthesis by specifying the maximum allowable norms for the complementary filters.
The proper design of these weighting functions is essential for the successful implementation of the proposed $\mathcal{H}_\infty$ synthesis approach.

Three key considerations should guide the design of weighting functions.
First, only proper and stable transfer functions should be employed.
Second, the order of the weighting functions should remain reasonably small to minimize computational costs associated with solving the optimization problem and to facilitate practical implementation of the filters (as the order of the synthesized filters equals the sum of the weighting functions' orders).
Third, the fundamental limitations imposed by the complementary property (eqref:eq:detail_control_sensor_comp_filter) must be respected, which implies that $|H_1(j\omega)|$ and $|H_2(j\omega)|$ cannot both be made small at the same frequency.

When designing complementary filters, it is typically desirable to specify their slopes, "blending" frequency, and maximum gains at low and high frequencies.
To facilitate the expression of these specifications, formula eqref:eq:detail_control_sensor_weight_formula is proposed for the design of weighting functions.
The parameters in this formula are $G_0 = \lim_{\omega \to 0} |W(j\omega)|$ (the low-frequency gain), $G_\infty = \lim_{\omega \to \infty} |W(j\omega)|$ (the high-frequency gain), $G_c = |W(j\omega_c)|$ (the gain at a specific frequency $\omega_c$ in $\si{rad/s}$), and $n$ (the slope between high and low frequency, which also corresponds to the order of the weighting function).
The typical magnitude response of a weighting function generated using eqref:eq:detail_control_sensor_weight_formula is illustrated in Figure ref:fig:detail_control_sensor_weight_formula.

#+begin_src matlab :exports none
%% Weighting Function Design
% Parameters
n = 3; w0 = 2*pi*10; G0 = 1e-3; G1 = 1e1; Gc = 2;

% Formulas
W = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (G0/Gc)^(1/n))/((1/G1)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (1/Gc)^(1/n)))^n;

% Function generateWF can be used to easily design the weighting filters
% W = generateWF('n', 3, 'w0', 2*pi*10, 'G0', 1e-3, 'Ginf', 10, 'Gc', 2);
#+end_src

#+begin_src matlab :exports none :results none
%% Magnitude of the weighting function with parameters
figure;
hold on;
plot(freqs, abs(squeeze(freqresp(W, freqs, 'Hz'))), 'k-');

plot([1e-3 1e0], [G0 G0], 'k--', 'LineWidth', 1)
text(1e0, G0, '$\quad G_0$')

plot([1e1 1e3], [G1 G1], 'k--', 'LineWidth', 1)
text(1e1,G1,'$G_{\infty}\quad$','HorizontalAlignment', 'right')

plot([w0/2/pi w0/2/pi], [1 2*Gc], 'k--', 'LineWidth', 1)
text(w0/2/pi,1,'$\omega_c$','VerticalAlignment', 'top', 'HorizontalAlignment', 'center')

plot([w0/2/pi/2  2*w0/2/pi], [Gc Gc], 'k--', 'LineWidth', 1)
text(w0/2/pi/2, Gc, '$G_c \quad$','HorizontalAlignment', 'right')

text(w0/5/pi/2, abs(evalfr(W, j*w0/5)), 'Slope: $n \quad$', 'HorizontalAlignment', 'right')

text(w0/2/pi, abs(evalfr(W, j*w0)), '$\bullet$', 'HorizontalAlignment', 'center')
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
hold off;
xlim([freqs(1), freqs(end)]);
ylim([5e-4, 20]);
yticks([1e-4, 1e-3, 1e-2, 1e-1, 1, 1e1]);
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/detail_control_sensor_weight_formula.pdf', 'width', 'half', 'height', 'normal');
#+end_src

#+attr_latex: :options []{0.49\linewidth}
#+begin_minipage
#+name: fig:detail_control_sensor_weight_formula
#+caption: Magnitude of a weighting function generated using eqref:eq:detail_control_sensor_weight_formula, $G_0 = 10^{-3}$, $G_\infty = 10$, $\omega_c = \SI{10}{Hz}$, $G_c = 2$, $n = 3$.
#+attr_latex: :width 0.95\linewidth :float nil
[[file:figs/detail_control_sensor_weight_formula.png]]
#+end_minipage
\hfill
#+attr_latex: :options []{0.49\linewidth}
#+begin_minipage
\begin{equation}\label{eq:detail_control_sensor_weight_formula}
  W(s) = \left( \frac{
           \hfill{} \frac{1}{\omega_c} \sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{1 - \left(\frac{G_c}{G_\infty}\right)^{\frac{2}{n}}}} s + \left(\frac{G_0}{G_c}\right)^{\frac{1}{n}}
         }{
           \left(\frac{1}{G_\infty}\right)^{\frac{1}{n}} \frac{1}{\omega_c} \sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{1 - \left(\frac{G_c}{G_\infty}\right)^{\frac{2}{n}}}} s + \left(\frac{1}{G_c}\right)^{\frac{1}{n}}
         }\right)^n
\end{equation}
#+end_minipage

**** Validation of the proposed synthesis method

The proposed methodology for designing complementary filters is now applied to a simple example.
Consider the design of two complementary filters $H_1(s)$ and $H_2(s)$ with the following requirements:
- The blending frequency should be approximately $\SI{10}{Hz}$
- The slope of $|H_1(j\omega)|$ should be $+2$ below $\SI{10}{Hz}$, with a low-frequency gain of $10^{-3}$
- The slope of $|H_2(j\omega)|$ should be $-3$ above $\SI{10}{Hz}$, with a high-frequency gain of $10^{-3}$

The first step involves translating these requirements by appropriately designing the weighting functions.
The formula proposed in eqref:eq:detail_control_sensor_weight_formula is employed for this purpose.
The parameters used are summarized in Table ref:tab:detail_control_sensor_weights_params.
The inverse magnitudes of the designed weighting functions, which represent the maximum allowable norms of the complementary filters, are depicted by the dashed lines in Figure ref:fig:detail_control_sensor_hinf_filters_results.

#+begin_src matlab
%% Synthesis of Complementary Filters using H-infinity synthesis
% Design of the Weighting Functions
W1 = generateWF('n', 3, 'w0', 2*pi*10, 'G0', 1000, 'Ginf', 1/10, 'Gc', 0.45);
W2 = generateWF('n', 2, 'w0', 2*pi*10, 'G0', 1/10, 'Ginf', 1000, 'Gc', 0.45);

% Generalized Plant
P = [W1 -W1;
     0   W2;
     1   0];

% H-Infinity Synthesis
[H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');

% Define H1 to be the complementary of H2
H1 = 1 - H2;

% The function generateCF can also be used to synthesize the complementary filters.
% [H1, H2] = generateCF(W1, W2);
#+end_src

#+begin_src matlab :exports none :results none
%% Bode plot of the Weighting filters and Obtained complementary filters
figure;
hold on;
plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'color', colors(1,:),'DisplayName', '$|W_1|^{-1}$');
plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'color', colors(2,:),'DisplayName', '$|W_2|^{-1}$');
plot(freqs,    abs(squeeze(freqresp(H1, freqs, 'Hz'))), '-',  'color', [colors(1,:), 0.5], 'linewidth', 2.5,'DisplayName', '$H_1$');
plot(freqs,    abs(squeeze(freqresp(H2, freqs, 'Hz'))), '-',  'color', [colors(2,:), 0.5], 'linewidth', 2.5,'DisplayName', '$H_2$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
ylim([8e-4, 20]);
xlim([freqs(1), freqs(end)]);
leg = legend('location', 'south', 'FontSize', 8, 'NumColumns', 2);
leg.ItemTokenSize(1) = 18;
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/detail_control_sensor_hinf_filters_results.pdf', 'width', 'half', 'height', 400);
#+end_src

#+attr_latex: :options [b]{0.44\linewidth}
#+begin_minipage
#+attr_latex: :environment tabularx :width 0.7\linewidth :placement [b] :align ccc
#+attr_latex: :booktabs t :float nil
| Parameter   |         $W_1(s)$ |         $W_2(s)$ |
|-------------+------------------+------------------|
| $G_0$       |              $0.1$ |             $1000$ |
| $G_{\infty}$ |             $1000$ |              $0.1$ |
| $\omega_c$  | $2 \pi \cdot 10$ | $2 \pi \cdot 10$ |
| $G_c$       |             $0.45$ |             $0.45$ |
| $n$         |                $2$ |                $3$ |
#+latex: \captionof{table}{\label{tab:detail_control_sensor_weights_params}Parameters for \(W_1(s)\) and \(W_2(s)\)}
#+end_minipage
\hfill
#+attr_latex: :options [b]{0.52\linewidth}
#+begin_minipage
#+name: fig:detail_control_sensor_hinf_filters_results
#+attr_latex: :scale 1 :float nil
#+caption: Weights and obtained filters
[[file:figs/detail_control_sensor_hinf_filters_results.png]]
#+end_minipage

Standard $\mathcal{H}_\infty$ synthesis is then applied to the generalized plant shown in Figure ref:fig:detail_control_sensor_h_infinity_robust_fusion_plant.
This yields the filter $H_2(s)$ that minimizes the $\mathcal{H}_\infty$ norm from input $w$ to outputs $[z_1,\ z_2]^{\intercal}$.
The resulting $\mathcal{H}_\infty$ norm is found to be close to unity, indicating successful synthesis: the norms of the complementary filters remain below the specified upper bounds.
This is confirmed by the Bode plots of the obtained complementary filters in Figure ref:fig:detail_control_sensor_hinf_filters_results.
This straightforward example demonstrates that the proposed methodology for shaping complementary filters is both simple and effective.

** Synthesis of a set of three complementary filters
<<ssec:detail_control_sensor_hinf_three_comp_filters>>

Some applications require merging more than two sensors [[cite:&stoten01_fusion_kinet_data_using_compos_filter;&fonseca15_compl]].
For instance, at LIGO, three sensors (an LVDT, a seismometer, and a geophone) are merged to form a super sensor [[cite:&matichard15_seism_isolat_advan_ligo]].

When merging $n>2$ sensors using complementary filters, two architectures can be employed as shown in Figure ref:fig:detail_control_sensor_fusion_three.
The fusion can be performed either in a "sequential" manner where $n-1$ sets of two complementary filters are used (Figure ref:fig:detail_control_sensor_fusion_three_sequential), or in a "parallel" manner where one set of $n$ complementary filters is used (Figure ref:fig:detail_control_sensor_fusion_three_parallel).

In the sequential approach, typical sensor fusion synthesis techniques can be applied.
However, when a parallel architecture is implemented, a new synthesis method for a set of more than two complementary filters is required, as only simple analytical formulas have been proposed in the literature [[cite:&stoten01_fusion_kinet_data_using_compos_filter;&fonseca15_compl]].
A generalization of the proposed complementary filter synthesis method is presented in this section.

#+begin_src latex :file detail_control_sensor_fusion_three_sequential.pdf
\tikzset{block/.default={0.8cm}{0.8cm}}
\tikzset{addb/.append style={scale=0.7}}
\tikzset{node distance=0.6}
\begin{tikzpicture}
  \node[branch] (x) at (0, 0);

  \node[block, right=0.4 of x] (sensor2) {Sensor 2};
  \node[block, above=0.4 of sensor2] (sensor1) {Sensor 1};
  \node[block, below=0.4 of sensor2] (sensor3) {Sensor 3};

  \node[block, right=1.1 of sensor1](H1){$H_1(s)$};
  \node[block, right=1.1 of sensor2](H2){$H_2(s)$};
  \node[addb] (add) at ($0.5*(H1.east)+0.5*(H2.east)+(0.6, 0)$){};

  \node[block, right=0.8 of add](H1p)  {$H_1^\prime(s)$};
  \node[block] (H2p) at (H1p|-sensor3) {$H_2^\prime(s)$};

  \node[addb] (addp) at ($0.5*(H1p.east)+0.5*(H2p.east)+(0.6, 0)$){};

  \draw[->] ($(x)+(-0.8, 0)$) node[above right]{$x$} -- (sensor2.west);
  \draw[->] (x.center)     |- (sensor1.west);
  \draw[->] (x.center)     |- (sensor3.west);
  \draw[->] (sensor1.east) -- (H1.west)  node[above left]{$\hat{x}_1$};
  \draw[->] (sensor2.east) -- (H2.west)  node[above left]{$\hat{x}_2$};
  \draw[->] (sensor3.east) -- (H2p.west) node[above left]{$\hat{x}_3$};
  \draw[->] (H1)           -| (add.north);
  \draw[->] (H2)           -| (add.south);
  \draw[->] (add.east)     -- (H1p.west) node[above left]{$\hat{x}_{12}$};
  \draw[->] (H1p)          -| (addp.north);
  \draw[->] (H2p)          -| (addp.south);
  \draw[->] (addp.east)    -- ++(0.8, 0) node[above left]{$\hat{x}$};

  \begin{scope}[on background layer]
    \node[fit={(x.west|-sensor3.south) (sensor1.north-|addp.east)}, fill=black!10!white, draw, inner sep=6pt] (supersensor) {};

    \node[fit={(x.west|-sensor1.north) (add.east|-sensor2.south)}, fill=black!20!white, draw, inner sep=3pt] (superinertialsensor) {};
  \end{scope}
\end{tikzpicture}
#+end_src

#+begin_src latex :file detail_control_sensor_fusion_three_parallel.pdf
\tikzset{block/.default={0.8cm}{0.8cm}}
\tikzset{addb/.append style={scale=0.7}}
\tikzset{node distance=0.6}
\begin{tikzpicture}
  \node[branch] (x) at (0, 0);
  \node[block, right=0.4 of x] (sensor2) {Sensor 2};
  \node[block, above=0.3 of sensor2] (sensor1) {Sensor 1};
  \node[block, below=0.3 of sensor2] (sensor3) {Sensor 3};

  \node[block, right=1.1 of sensor1](H1){$H_1(s)$};
  \node[block, right=1.1 of sensor2](H2){$H_2(s)$};
  \node[block, right=1.1 of sensor3](H3){$H_3(s)$};

  \node[addb, right=0.6 of H2](add){};

  \draw[->] (x.center)                             |- (sensor1.west);
  \draw[] ($(x)+(-0.8, 0)$) node[above right]{$x$} -- (sensor2.west);
  \draw[->] (x.center)                             |- (sensor3.west);

  \draw[->] (sensor1.east) -- (H1.west) node[above left]{$\hat{x}_1$};
  \draw[->] (sensor2.east) -- (H2.west) node[above left]{$\hat{x}_2$};
  \draw[->] (sensor3.east) -- (H3.west) node[above left]{$\hat{x}_3$};

  \draw[->] (H1) -| (add.north);
  \draw[->] (H2) -- (add.west);
  \draw[->] (H3) -| (add.south);

  \draw[->] (add.east) -- ++(0.8, 0) node[above left]{$\hat{x}$};

  \begin{scope}[on background layer]
    \node[fit={(H3.south-|x) (H1.north-|add.east)}, fill=black!10!white, draw, inner sep=6pt] (supersensor) {};
  \end{scope}
\end{tikzpicture}
#+end_src

#+name: fig:detail_control_sensor_fusion_three
#+caption: Possible sensor fusion architecture when more than two sensors are to be merged
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_fusion_three_sequential}Sequential fusion}
#+attr_latex: :options {0.58\textwidth}
#+begin_subfigure
#+attr_latex: :scale 0.9
[[file:figs/detail_control_sensor_fusion_three_sequential.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_fusion_three_parallel}Parallel fusion}
#+attr_latex: :options {0.38\textwidth}
#+begin_subfigure
#+attr_latex: :scale 0.9
[[file:figs/detail_control_sensor_fusion_three_parallel.png]]
#+end_subfigure
#+end_figure

The synthesis objective is to compute a set of $n$ stable transfer functions $[H_1(s),\ H_2(s),\ \dots,\ H_n(s)]$ that satisfy conditions eqref:eq:detail_control_sensor_hinf_cond_compl_gen and eqref:eq:detail_control_sensor_hinf_cond_perf_gen.

\begin{subequations}\label{eq:detail_control_sensor_hinf_problem_gen}
  \begin{align}
    & \sum_{i=1}^n H_i(s) = 1 \label{eq:detail_control_sensor_hinf_cond_compl_gen} \\
    & \left| H_i(j\omega) \right| < \frac{1}{\left| W_i(j\omega) \right|}, \quad \forall \omega,\ i = 1 \dots n \label{eq:detail_control_sensor_hinf_cond_perf_gen}
  \end{align}
\end{subequations}

The transfer functions $[W_1(s),\ W_2(s),\ \dots,\ W_n(s)]$ are weights selected to specify the maximum complementary filters' norm during synthesis.

This synthesis objective is closely related to the one described in Section ref:ssec:detail_control_sensor_hinf_method, and the proposed synthesis method represents a generalization of the approach previously presented.
A set of $n$ complementary filters can be shaped by applying standard $\mathcal{H}_\infty$ synthesis to the generalized plant $P_n(s)$ described by eqref:eq:detail_control_sensor_generalized_plant_n_filters.

\begin{equation}\label{eq:detail_control_sensor_generalized_plant_n_filters}
  \begin{bmatrix} z_1 \\ \vdots \\ z_n \\ v \end{bmatrix} = P_n(s) \begin{bmatrix} w \\ u_1 \\ \vdots \\ u_{n-1} \end{bmatrix}; \quad
  P_n(s) = \begin{bmatrix}
    W_1    & -W_1   & \dots  & \dots  & -W_1 \\
    0      &  W_2   & 0      & \dots  & 0 \\
    \vdots & \ddots & \ddots & \ddots & \vdots \\
    \vdots &        & \ddots & \ddots & 0 \\
    0      & \dots  & \dots  & 0      & W_n \\
    1      & 0      & \dots  & \dots  & 0
    \end{bmatrix}
\end{equation}

If the synthesis is successful, a set of $n-1$ filters $[H_2(s),\ H_3(s),\ \dots,\ H_n(s)]$ is obtained such that eqref:eq:detail_control_sensor_hinf_syn_obj_gen is satisfied.

\begin{equation}\label{eq:detail_control_sensor_hinf_syn_obj_gen}
  \left\|\begin{matrix} \left(1 - \left[ H_2(s) + H_3(s) + \dots + H_n(s) \right]\right) W_1(s) \\ H_2(s) W_2(s) \\ \vdots \\ H_n(s) W_n(s) \end{matrix}\right\|_\infty \le 1
\end{equation}

$H_1(s)$ is then defined using eqref:eq:detail_control_sensor_h1_comp_h2_hn, which ensures the complementary property for the set of $n$ filters eqref:eq:detail_control_sensor_hinf_cond_compl_gen.
Condition eqref:eq:detail_control_sensor_hinf_cond_perf_gen is satisfied through eqref:eq:detail_control_sensor_hinf_syn_obj_gen.

\begin{equation}\label{eq:detail_control_sensor_h1_comp_h2_hn}
  H_1(s) \triangleq 1 - \big[ H_2(s) + H_3(s) + \dots + H_n(s) \big]
\end{equation}

To validate the proposed method for synthesizing a set of three complementary filters, an example is provided.
The sensors to be merged are a displacement sensor (effective from DC up to $\SI{1}{Hz}$), a geophone (effective from $1$ to $\SI{10}{Hz}$), and an accelerometer (effective above $\SI{10}{Hz}$).
Three weighting functions are designed using formula eqref:eq:detail_control_sensor_weight_formula, and their inverse magnitudes are shown in Figure ref:fig:detail_control_sensor_three_complementary_filters_results (dashed curves).

Consider the generalized plant $P_3(s)$ shown in Figure ref:fig:detail_control_sensor_comp_filter_three_hinf_fb, which is also described by eqref:eq:detail_control_sensor_generalized_plant_three_filters.

\begin{equation}\label{eq:detail_control_sensor_generalized_plant_three_filters}
  \begin{bmatrix} z_1 \\ z_2 \\ z_3 \\ v \end{bmatrix} = P_3(s) \begin{bmatrix} w \\ u_1 \\ u_2 \end{bmatrix}; \quad P_3(s) = \begin{bmatrix}W_1(s) & -W_1(s) & -W_1(s) \\ 0 & \phantom{+}W_2(s) & 0 \\ 0 & 0 & \phantom{+}W_3(s) \\ 1 & 0 & 0 \end{bmatrix}
\end{equation}

#+begin_src latex :file detail_control_sensor_comp_filter_three_hinf_gen_plant.pdf
\tikzset{block/.default={0.8cm}{0.8cm}}
\tikzset{addb/.append style={scale=0.7}}
\tikzset{node distance=0.6}
\begin{tikzpicture}
  \node[block={5.0cm}{4.5cm}, fill=black!10!white] (P) {};
  \node[above] at (P.north) {$P_3(s)$};

  \coordinate[] (inputw)  at ($(P.south west)!0.8!(P.north west) + (-0.7, 0)$);
  \coordinate[] (inputu)  at ($(P.south west)!0.4!(P.north west) + (-0.7, 0)$);

  \coordinate[] (output1) at ($(P.south east)!0.8!(P.north east)  + (0.7, 0)$);
  \coordinate[] (output2) at ($(P.south east)!0.55!(P.north east) + (0.7, 0)$);
  \coordinate[] (output3) at ($(P.south east)!0.3!(P.north east)  + (0.7, 0)$);
  \coordinate[] (outputv) at ($(P.south east)!0.1!(P.north east)  + (0.7, 0)$);

  \node[block, left=1.4 of output1] (W1){$W_1(s)$};
  \node[block, left=1.4 of output2] (W2){$W_2(s)$};
  \node[block, left=1.4 of output3] (W3){$W_3(s)$};
  \node[addb={+}{}{}{}{-}, left=of W1] (sub1) {};
  \node[addb={+}{}{}{}{-}, left=of sub1] (sub2) {};

  \node[block, below=0.3 of P, opacity=0] (H) {$\begin{bmatrix}H_2(s) \\ H_3(s)\end{bmatrix}$};

  \draw[->] (inputw) node[above right](w){$w$} -- (sub2.west);
  \draw[->] (W3-|sub1)node[branch]{} -- (sub1.south);
  \draw[->] (W2-|sub2)node[branch]{} -- (sub2.south);
  \draw[->] ($(sub2.west)+(-0.5, 0)$) node[branch]{} |- (outputv);
  \draw[->] (inputu|-W2) -- (W2.west);
  \draw[->] (inputu|-W3) -- (W3.west);

  \draw[->] (sub2.east) -- (sub1.west);
  \draw[->] (sub1.east) -- (W1.west);
  \draw[->] (W1.east) -- (output1)node[above left](z){$z_1$};
  \draw[->] (W2.east) -- (output2)node[above left]{$z_2$};
  \draw[->] (W3.east) -- (output3)node[above left]{$z_3$};
  \node[above] at (W2-|w){$u_1$};
  \node[above] at (W3-|w){$u_2$};
  \node[above] at (outputv-|z){$v$};
\end{tikzpicture}
#+end_src

#+begin_src latex :file detail_control_sensor_comp_filter_three_hinf_fb.pdf
\tikzset{block/.default={0.8cm}{0.8cm}}
\tikzset{addb/.append style={scale=0.7}}
\tikzset{node distance=0.6}
\begin{tikzpicture}
  \node[block={5.0cm}{4.5cm}, fill=black!10!white] (P) {};
  \node[above] at (P.north) {$P_3(s)$};

  \coordinate[] (inputw)  at ($(P.south west)!0.8!(P.north west) + (-0.7, 0)$);
  \coordinate[] (inputu)  at ($(P.south west)!0.4!(P.north west) + (-0.7, 0)$);

  \coordinate[] (output1) at ($(P.south east)!0.8!(P.north east)  + (0.7, 0)$);
  \coordinate[] (output2) at ($(P.south east)!0.55!(P.north east) + (0.7, 0)$);
  \coordinate[] (output3) at ($(P.south east)!0.3!(P.north east)  + (0.7, 0)$);
  \coordinate[] (outputv) at ($(P.south east)!0.1!(P.north east)  + (0.7, 0)$);

  \node[block, left=1.4 of output1] (W1){$W_1(s)$};
  \node[block, left=1.4 of output2] (W2){$W_2(s)$};
  \node[block, left=1.4 of output3] (W3){$W_3(s)$};
  \node[addb={+}{}{}{}{-}, left=of W1] (sub1) {};
  \node[addb={+}{}{}{}{-}, left=of sub1] (sub2) {};

  \node[block, below=0.3 of P] (H) {$\begin{bmatrix}H_2(s) \\ H_3(s)\end{bmatrix}$};

  \draw[->] (inputw) node[above right](w){$w$} -- (sub2.west);
  \draw[->] (W3-|sub1)node[branch]{} -- (sub1.south);
  \draw[->] (W2-|sub2)node[branch]{} -- (sub2.south);
  \draw[->] ($(sub2.west)+(-0.5, 0)$) node[branch]{} |- (outputv) |- (H.east);
  \draw[->] ($(H.south west)!0.7!(H.north west)$) -| (inputu|-W2) -- (W2.west);
  \draw[->] ($(H.south west)!0.3!(H.north west)$) -| ($(inputu|-W3)+(0.4, 0)$) -- (W3.west);

  \draw[->] (sub2.east) -- (sub1.west);
  \draw[->] (sub1.east) -- (W1.west);
  \draw[->] (W1.east) -- (output1)node[above left](z){$z_1$};
  \draw[->] (W2.east) -- (output2)node[above left]{$z_2$};
  \draw[->] (W3.east) -- (output3)node[above left]{$z_3$};
  \node[above] at (W2-|w){$u_1$};
  \node[above] at (W3-|w){$u_2$};
  \node[above] at (outputv-|z){$v$};
\end{tikzpicture}
#+end_src

#+name: fig:detail_control_sensor_comp_filter_three_hinf
#+caption: Architecture for the $\mathcal{H}_\infty$ synthesis of three complementary filters (\subref{fig:detail_control_sensor_comp_filter_three_hinf_fb}). Bode plot of the inverse weighting functions and of the three obtained complementary filters (\subref{fig:detail_control_sensor_three_complementary_filters_results})
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_comp_filter_three_hinf_fb}Generalized plant with the synthesized filter}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
#+attr_latex: :width 0.95\linewidth
[[file:figs/detail_control_sensor_comp_filter_three_hinf_fb.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_three_complementary_filters_results}Weights and obtained filters}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
#+attr_latex: :width 0.95\linewidth
[[file:figs/detail_control_sensor_three_complementary_filters_results.png]]
#+end_subfigure
#+end_figure

Standard $\mathcal{H}_\infty$ synthesis is performed on the generalized plant $P_3(s)$.
Two filters, $H_2(s)$ and $H_3(s)$, are obtained such that the $\mathcal{H}_\infty$ norm of the closed-loop transfer from $w$ to $[z_1,\ z_2,\ z_3]$ of the system in Figure ref:fig:detail_control_sensor_comp_filter_three_hinf_fb is less than one.
Filter $H_1(s)$ is defined using eqref:eq:detail_control_sensor_h1_compl_h2_h3, thus ensuring the complementary property of the obtained set of filters.

\begin{equation}\label{eq:detail_control_sensor_h1_compl_h2_h3}
  H_1(s) \triangleq 1 - \big[ H_2(s) + H_3(s) \big]
\end{equation}

Figure ref:fig:detail_control_sensor_three_complementary_filters_results displays the three synthesized complementary filters (solid lines), confirming the successful synthesis.

#+begin_src matlab
%% Synthesis of a set of three complementary filters
% Design of the Weighting Functions
W1 = generateWF('n', 2, 'w0', 2*pi*1, 'G0', 1/10, 'Ginf', 1000, 'Gc', 0.5);
W2 = 0.22*(1 + s/2/pi/1)^2/(sqrt(1e-4) + s/2/pi/1)^2*(1 + s/2/pi/10)^2/(1 + s/2/pi/1000)^2;
W3 = generateWF('n', 3, 'w0', 2*pi*10, 'G0', 1000, 'Ginf', 1/10, 'Gc', 0.5);

% Generalized plant for the synthesis of 3 complementary filters
P = [W1 -W1 -W1;
     0   W2  0 ;
     0   0   W3;
     1   0   0];

% Standard H-Infinity Synthesis
[H, ~, gamma, ~] = hinfsyn(P, 1, 2,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');

% Synthesized H2 and H3 filters
H2 = tf(H(1));
H3 = tf(H(2));

% H1 is defined as the complementary filter of H2 and H3
H1 = 1 - H2 - H3;
#+end_src

#+begin_src matlab :exports none :results none
%% Bode plot of the inverse weighting functions and of the three complementary filters obtained using the H-infinity synthesis
figure;
hold on;
plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'color', colors(1,:),'DisplayName', '$|W_1|^{-1}$');
plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'color', colors(2,:),'DisplayName', '$|W_2|^{-1}$');
plot(freqs, 1./abs(squeeze(freqresp(W3, freqs, 'Hz'))), '--', 'color', colors(3,:),'DisplayName', '$|W_3|^{-1}$');
plot(freqs,    abs(squeeze(freqresp(H1, freqs, 'Hz'))), '-',  'color', [colors(1,:), 0.5], 'linewidth', 2.5,'DisplayName', '$H_1$');
plot(freqs,    abs(squeeze(freqresp(H2, freqs, 'Hz'))), '-',  'color', [colors(2,:), 0.5], 'linewidth', 2.5,'DisplayName', '$H_2$');
plot(freqs,    abs(squeeze(freqresp(H3, freqs, 'Hz'))), '-',  'color', [colors(3,:), 0.5], 'linewidth', 2.5,'DisplayName', '$H_3$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
xlim([freqs(1), freqs(end)]); ylim([1e-4, 20]);
leg = legend('location', 'southeast', 'FontSize', 8);
leg.ItemTokenSize(1) = 18;
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/detail_control_sensor_three_complementary_filters_results.pdf', 'width', 'half', 'height', 'normal');
#+end_src

** Conclusion
:PROPERTIES:
:UNNUMBERED: t
:END:

A new method for designing complementary filters using the $\mathcal{H}_\infty$ synthesis has been proposed.
This approach allows shaping of the filter magnitudes through the use of weighting functions during synthesis.
This capability is particularly valuable in practice since the characteristics of the super sensor are directly linked to the complementary filters' magnitude.
Consequently, typical sensor fusion objectives can be effectively translated into requirements on the magnitudes of the filters.

For the Nano Active Stabilization System (NASS), the High Authority Control-Integral Force Feedback (HAC-IFF) strategy was found to perform well and to offer the advantages of being both intuitive to understand and straightforward to tune.
Looking forward, it would be interesting to investigate how sensor fusion (particularly between the force sensor and external metrology) compares to the HAC-IFF approach in terms of performance and robustness.

* Decoupling
:PROPERTIES:
:HEADER-ARGS:matlab+: :tangle matlab/detail_control_2_decoupling.m
:END:
<<sec:detail_control_decoupling>>

# file:~/Cloud/research/matlab/decoupling-strategies/svd-control.org

** Introduction                                                      :ignore:

When dealing with MIMO systems, a typical strategy is to:
- First decouple the plant dynamics (discussed in this section)
- Apply SISO control for the decoupled plant (discussed in section ref:sec:detail_control_cf)

Another strategy would be to apply a multivariable control synthesis to the coupled system.
Strangely, while H-infinity synthesis is a mature technology, it use for the control of Stewart platform is not yet demonstrated.
From [[cite:&thayer02_six_axis_vibrat_isolat_system]]:
#+begin_quote
Experimental closed-loop control results using the hexapod have shown that controllers designed using a decentralized single-strut design work well when compared to full multivariable methodologies.
#+end_quote

- [X] Review of [[file:~/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/bibliography.org::*Decoupling Strategies][Decoupling Strategies]] for stewart platforms
- [ ] Add some citations about different methods
- [ ] Maybe transform table into text

#+name: tab:detail_control_decoupling_review
#+caption: Litterature review about decoupling strategy for Stewart platform control
#+attr_latex: :environment tabularx :width 0.9\linewidth :align Xccc
#+attr_latex: :center t :booktabs t :font \scriptsize
| *Actuators*   | *Sensors*                                          | *Control*                                                                             | *Reference*                                                                                                            |
|---------------+----------------------------------------------------+---------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------|
| APA           | Eddy current displacement                          | *Decentralized* (struts) PI + LPF control                                             | [[cite:&furutani04_nanom_cuttin_machin_using_stewar]]                                                                      |
| PZT Piezo     | Strain Gauge                                       | Decentralized position feedback                                                       | [[cite:&du14_piezo_actuat_high_precis_flexib]]                                                                             |
|---------------+----------------------------------------------------+---------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------|
| Voice Coil    | Force                                              | *Cartesian frame* decoupling                                                          | [[cite:&obrien98_lesson]]                                                                                                  |
| Voice Coil    | Force                                              | Cartesian Frame, Jacobians, IFF                                                       | [[cite:&mcinroy99_dynam;&mcinroy99_precis_fault_toler_point_using_stewar_platf;&mcinroy00_desig_contr_flexur_joint_hexap]] |
| Hydraulic     | LVDT                                               | Decentralized (strut) vs Centralized (cartesian)                                      | [[cite:&kim00_robus_track_contr_desig_dof_paral_manip]]                                                                    |
| Voice Coil    | Accelerometer (collocated), ext. Rx/Ry sensors     | Cartesian acceleration feedback (isolation) + 2DoF pointing control (external sensor) | [[cite:&li01_simul_vibrat_isolat_point_contr]]                                                                             |
| Voice Coil    | Accelerometer in each leg                          | Centralized Vibration Control, PI, Skyhook                                            | [[cite:&abbas14_vibrat_stewar_platf]]                                                                                      |
|---------------+----------------------------------------------------+---------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------|
| Voice Coil    | Geophone + Eddy Current (Struts, collocated)       | Decentralized (Sky Hook) + Centralized (*modal*) Control                              | [[cite:&pu11_six_degree_of_freed_activ]]                                                                                   |
| Piezoelectric | Force, Position                                    | Vibration isolation, Model-Based, *Modal control*: 6x PI controllers                  | [[cite:&yang19_dynam_model_decoup_contr_flexib]]                                                                           |
|---------------+----------------------------------------------------+---------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------|
| PZT           | Geophone (struts)                                  | *H-Infinity* and mu-synthesis                                                         | [[cite:&lei08_multi_objec_robus_activ_vibrat]]                                                                             |
| Voice Coil    | Force sensors (struts) + accelerometer (cartesian) | Decentralized Force Feedback + Centralized H2 control based on accelerometers         | [[cite:&xie17_model_contr_hybrid_passiv_activ]]                                                                            |
| Voice Coil    | Accelerometers                                     | MIMO H-Infinity, active damping                                                       | [[cite:&jiao18_dynam_model_exper_analy_stewar]]                                                                            |


The goal of this section is to compare the use of several methods for the decoupling of parallel manipulators.

It is structured as follow:
- Section ref:ssec:detail_control_decoupling_model: the model used to compare/test decoupling strategies is presented
- Section ref:ssec:detail_control_decoupling_jacobian: decoupling using Jacobian matrices is presented
- Section ref:ssec:detail_control_decoupling_modal: modal decoupling is presented
- Section ref:ssec:detail_control_decoupling_svd: SVD decoupling is presented
- Section ref:ssec:detail_control_decoupling_comp: the three decoupling methods are applied on the test model and compared
- Conclusions are drawn on the three decoupling methods

** 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

#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src

#+begin_src matlab :tangle no :noweb yes
<<m-init-path>>
#+end_src

#+begin_src matlab :eval no :noweb yes
<<m-init-path-tangle>>
#+end_src

#+begin_src matlab :noweb yes
<<m-init-other>>

%% Initialize Frequency Vector
freqs = logspace(0, 3, 1000);
#+end_src

** Test Model
<<ssec:detail_control_decoupling_model>>

- Instead of comparing the decoupling strategies using the Stewart platform, a similar yet much simpler parallel manipulator is used instead
- to render the analysis simpler, the system of Figure ref:fig:detail_control_decoupling_model_details is used
- Fully parallel manipulator: it has 3DoF, and has 3 parallels struts whose model is shown in Figure ref:fig:detail_control_decoupling_strut_model
  As many DoF as actuators and sensors
- It is quite similar to the Stewart platform (parallel architecture, as many struts as DoF)

Two frames are defined:
- $\{M\}$ with origin $O_M$ at the Center of mass of the solid body
- $\{K\}$ with origin $O_K$ at the Center of mass of the parallel manipulator

#+name: fig:detail_control_decoupling_model_details
#+caption: 3DoF model used to study decoupling strategies
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:detail_control_decoupling_model_test}Geometrical parameters}
#+attr_latex: :options {0.58\textwidth}
#+begin_subfigure
#+attr_latex: :scale 1
[[file:figs/detail_control_decoupling_model_test.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_control_decoupling_strut_model}Strut model}
#+attr_latex: :options {0.38\textwidth}
#+begin_subfigure
#+attr_latex: :scale 1
[[file:figs/detail_control_decoupling_strut_model.png]]
#+end_subfigure
#+end_figure

First, the equation of motion are derived.
Expressing the second law of Newton on the suspended mass, expressed at its center of mass gives

\begin{equation}
  \bm{M}_{\{M\}} \ddot{\bm{\mathcal{X}}}_{\{M\}}(t) = \sum \bm{\mathcal{F}}_{\{M\}}(t)
\end{equation}

with $\bm{\mathcal{X}}_{\{M\}}$ the two translation and one rotation expressed with respect to the center of mass and $\bm{\mathcal{F}}_{\{M\}}$ forces and torque applied at the center of mass.

\begin{equation}
  \bm{\mathcal{X}}_{\{M\}} = \begin{bmatrix}
    x \\
    y \\
    R_z
  \end{bmatrix}, \quad \bm{\mathcal{F}}_{\{M\}} = \begin{bmatrix}
    F_x \\
    F_y \\
    M_z
  \end{bmatrix}
\end{equation}

In order to map the spring, damping and actuator forces to XY forces and Z torque expressed at the center of mass, the Jacobian matrix $\bm{J}_{\{M\}}$ is used.

\begin{equation}\label{eq:detail_control_decoupling_jacobian_CoM}
  \bm{J}_{\{M\}} = \begin{bmatrix}
    1 & 0 &  h_a \\
    0 & 1 & -l_a \\
    0 & 1 &  l_a \\
  \end{bmatrix}
\end{equation}

Then, the equation of motion linking the actuator forces $\tau$ to the motion of the mass $\bm{\mathcal{X}}_{\{M\}}$ is obtained.

\begin{equation}\label{eq:detail_control_decoupling_plant_cartesian}
  \bm{M}_{\{M\}} \ddot{\bm{\mathcal{X}}}_{\{M\}}(t) + \bm{J}_{\{M\}}^{\intercal} \bm{\mathcal{C}} \bm{J}_{\{M\}} \dot{\bm{\mathcal{X}}}_{\{M\}}(t) + \bm{J}_{\{M\}}^{\intercal} \bm{\mathcal{K}} \bm{J}_{\{M\}} \bm{\mathcal{X}}_{\{M\}}(t) = \bm{J}_{\{M\}}^{\intercal} \bm{\tau}(t)
\end{equation}

Matrices representing the payload inertia as well as the actuator stiffness and damping are shown in

\begin{equation}
  \bm{M}_{\{M\}} = \begin{bmatrix}
    m & 0 & 0 \\
    0 & m & 0 \\
    0 & 0 & I
  \end{bmatrix}, \quad
  \bm{\mathcal{K}} = \begin{bmatrix}
    k & 0 & 0 \\
    0 & k & 0 \\
    0 & 0 & k
  \end{bmatrix}, \quad
  \bm{\mathcal{C}} = \begin{bmatrix}
    c & 0 & 0 \\
    0 & c & 0 \\
    0 & 0 & c
  \end{bmatrix}
\end{equation}

Parameters used for the following analysis are summarized in table ref:tab:detail_control_decoupling_test_model_params.

#+name: tab:detail_control_decoupling_test_model_params
#+caption: Model parameters
#+attr_latex: :environment tabularx :width 0.9\linewidth :align cXc
#+attr_latex: :center t :booktabs t :font \scriptsize
| *Parameter* | *Description*              | *Value*           |
|-------------+----------------------------+-------------------|
| $l_a$       |                            | $0.5\,m$          |
| $h_a$       |                            | $0.2\,m$          |
| $k$         | Actuator stiffness         | $10\,N/\mu m$     |
| $c$         | Actuator damping           | $200\,Ns/m$       |
| $m$         | Payload mass               | $40\,\text{kg}$   |
| $I$         | Payload rotational inertia | $5\,\text{kg}m^2$ |

** Control in the frame of the struts
<<ssec:detail_control_decoupling_decentralized>>

Let's first study the obtained dynamics in the frame of the struts.
The equation of motion linking actuator forces $\bm{\mathcal{\tau}}$ to strut relative motion $\bm{\mathcal{L}}$ is obtained from eqref:eq:detail_control_decoupling_plant_cartesian by mapping the cartesian motion of the mass to the relative motion of the struts using the Jacobian matrix $\bm{J}_{\{M\}}$ eqref:eq:detail_control_decoupling_jacobian_CoM .

The transfer function from $\bm{\mathcal{\tau}}$ to $\bm{\mathcal{L}}$ is shown in equation eqref:eq:detail_control_decoupling_plant_decentralized.

#+begin_src latex :file detail_control_decoupling_control_struts.pdf
\begin{tikzpicture}
  \node[block] (Gl) {$\bm{G}_{\{\mathcal{L}\}}$};

  % Connections and labels
  \draw[<-] (Gl.west) -- ++(-0.8, 0) node[above right]{$\bm{\tau}$};
  \draw[->] (Gl.east) -- ++( 0.8, 0) node[above left]{$\bm{\mathcal{L}}$};
\end{tikzpicture}
#+end_src

#+RESULTS:
[[file:figs/detail_control_decoupling_control_struts.png]]

\begin{equation}\label{eq:detail_control_decoupling_plant_decentralized}
  \frac{\bm{\mathcal{L}}}{\bm{\mathcal{\tau}}}(s) = \bm{G}_{\mathcal{L}}(s) = \left( \bm{J}_{\{M\}}^{-\intercal} \bm{M}_{\{M\}} \bm{J}_{\{M\}}^{-1} s^2 + \bm{\mathcal{C}} s + \bm{\mathcal{K}} \right)^{-1}
\end{equation}

At low frequency the plant converges to a diagonal constant matrix whose diagonal elements are linked to the actuator stiffnesses eqref:eq:detail_control_decoupling_plant_decentralized_low_freq.

\begin{equation}\label{eq:detail_control_decoupling_plant_decentralized_low_freq}
  \bm{G}_{\mathcal{L}}(j\omega) \xrightarrow[\omega \to 0]{} \bm{\mathcal{K}^{-1}}
\end{equation}

At high frequency, the plant converges to the mass matrix mapped in the frame of the struts, which is in general highly non-diagonal.

#+begin_src matlab
%% Compute Equation of motion
l = 1; h=2;
la = 0.5; % Horizontal position of actuators [m]
ha = 0.2; % Vertical of actuators [m]

m = 40; % Payload mass [kg]
I = 5; % Payload rotational inertia [kg m^2]

c = 2e2; % Actuator Damping [N/(m/s)]
k = 1e6; % Actuator Stiffness [N/m]

% Unit vectors of the actuators
s1 = [1;0];
s2 = [0;1];
s3 = [0;1];

% Stiffnesss and Damping matrices of the struts
Kr = diag([k,k,k]);
Cr = diag([c,c,c]);

%  Location of the joints with respect to the center of mass
Mb1 = [-l/2;-ha];
Mb2 = [-la; -h/2];
Mb3 = [ la; -h/2];

% Jacobian matrix (Center of Mass)
J_CoM = [s1', Mb1(1)*s1(2)-Mb1(2)*s1(1);
         s2', Mb2(1)*s2(2)-Mb2(2)*s2(1);
         s3', Mb3(1)*s3(2)-Mb3(2)*s3(1)];

% Mass Matrix in frame {M}
M = diag([m,m,I]);

% Stiffness Matrix in frame {M}
K = J_CoM'*Kr*J_CoM;

% Damping Matrix in frame {M}
C = J_CoM'*Cr*J_CoM;

% Plant in the frame of the struts
G_L = J_CoM*inv(M*s^2 + C*s + K)*J_CoM';
#+end_src

The magnitude of the coupled plant $\bm{G}_{\mathcal{L}}$ is shown in Figure ref:fig:detail_control_decoupling_coupled_plant_bode.
This confirms that at low frequency (below the first suspension mode), the plant is well decoupled.
Depending on the symmetry in the system, some diagonal elements may be equal (such as for struts 2 and 3 in this example).

#+begin_src matlab :exports none
figure;
tiledlayout(3, 3, 'TileSpacing', 'Compact', 'Padding', 'None');

for out_i = 1:3
    for in_i = 1:3
        nexttile;
        plot(freqs, abs(squeeze(freqresp(G_L(out_i,in_i), freqs, 'Hz'))), 'k-', ...
             'DisplayName', sprintf('$\\mathcal{L}_%i/\\tau_%i$', out_i, in_i));
        set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
        xlim([freqs(1), freqs(end)]); ylim([1e-8, 1e-4]);
        leg = legend('location', 'northeast', 'FontSize', 8);
        leg.ItemTokenSize(1) = 18;

        if in_i == 1
            ylabel('Mag. [m/N]')
        else
            set(gca, 'YTickLabel',[]);
        end

        if out_i == 3
            xlabel('Frequency [Hz]')
        else
            set(gca, 'XTickLabel',[]);
        end
    end
end
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/detail_control_decoupling_coupled_plant_bode.pdf', 'width', 'full', 'height', 'tall');
#+end_src

#+name: fig:detail_control_decoupling_coupled_plant_bode
#+caption: Magnitude of the coupled plant.
#+RESULTS:
[[file:figs/detail_control_decoupling_coupled_plant_bode.png]]

** Jacobian Decoupling
<<ssec:detail_control_decoupling_jacobian>>
**** Jacobian Matrix

As already explained, the Jacobian matrix can be used to both convert strut velocity $\dot{\mathcal{L}}$ to payload velocity and angular velocity $\dot{\bm{\mathcal{X}}}_{\{O\}}$ and Convert actuators forces $\bm{\tau}$ to forces/torque applied on the payload $\bm{\mathcal{F}}_{\{O\}}$ eqref:eq:detail_control_decoupling_jacobian.

\begin{subequations}\label{eq:detail_control_decoupling_jacobian}
  \begin{align}
    \dot{\bm{\mathcal{X}}}_{\{O\}} &= \bm{J}_{\{O\}} \dot{\bm{\mathcal{L}}}, \quad \dot{\bm{\mathcal{L}}} = \bm{J}_{\{O\}}^{-1} \dot{\bm{\mathcal{X}}}_{\{O\}} \\
    \bm{\mathcal{F}}_{\{O\}}       &= \bm{J}_{\{O\}}^{\intercal} \bm{\tau},            \quad \bm{\tau} = \bm{J}_{\{O\}}^{-\intercal} \bm{\mathcal{F}}_{\{O\}}
  \end{align}
\end{subequations}

The obtained plan (Figure ref:fig:detail_control_jacobian_decoupling_arch) has inputs and outputs that have physical meaning:
- $\bm{\mathcal{F}}_{\{O\}}$ are forces/torques applied on the payload at the origin of frame $\{O\}$
- $\bm{\mathcal{X}}_{\{O\}}$ are translations/rotation of the payload expressed in frame $\{O\}$

#+begin_src latex :file detail_control_decoupling_control_jacobian.pdf
\begin{tikzpicture}
  \node[block] (G) {$\bm{G}_{\{\mathcal{L}\}}$};
  \node[block, left=0.6 of G] (Jt) {$\bm{J}_{\{O\}}^{-\intercal}$};
  \node[block, right=0.6 of G] (Ja) {$\bm{J}_{\{O\}}^{-1}$};

  % Connections and labels
  \draw[<-] (Jt.west) -- ++(-1.4, 0) node[above right]{$\bm{\mathcal{F}}_{\{O\}}$};
  \draw[->] (Jt.east) -- (G.west)  node[above left]{$\bm{\tau}$};
  \draw[->] (G.east) -- (Ja.west)  node[above left]{$\bm{\mathcal{L}}$};
  \draw[->] (Ja.east) -- ++( 1.4, 0)  node[above left]{$\bm{\mathcal{X}}_{\{O\}}$};

  \begin{scope}[on background layer]
    \node[fit={(Jt.south west) (Ja.north east)}, fill=black!10!white, draw, dashed, inner sep=4pt] (Gx) {};
    \node[above] at (Gx.north) {$\bm{G}_{\{O\}}$};
  \end{scope}
\end{tikzpicture}
#+end_src

#+name: fig:detail_control_jacobian_decoupling_arch
#+caption: Block diagram of the transfer function from $\bm{\mathcal{F}}_{\{O\}}$ to $\bm{\mathcal{X}}_{\{O\}}$
#+RESULTS:
[[file:figs/detail_control_decoupling_control_jacobian.png]]

\begin{equation}\label{eq:detail_control_decoupling_plant_jacobian}
  \frac{\bm{\mathcal{X}}_{\{O\}}}{\bm{\mathcal{F}}_{\{O\}}}(s) = \bm{G}_{\{O\}}(s) = \left( \bm{J}_{\{O\}}^{\intercal} \bm{J}_{\{M\}}^{-\intercal} \bm{M}_{\{M\}} \bm{J}_{\{M\}}^{-1} \bm{J}_{\{O\}} s^2 + \bm{J}_{\{O\}}^{\intercal} \bm{\mathcal{C}} \bm{J}_{\{O\}} s + \bm{J}_{\{O\}}^{\intercal} \bm{\mathcal{K}} \bm{J}_{\{O\}} \right)^{-1}
\end{equation}

The frame $\{O\}$ can be any chosen frame, but the decoupling properties depends on the chosen frame $\{O\}$.
There are two natural choices: the center of mass $\{M\}$ and the center of stiffness $\{K\}$.
Note that the Jacobian matrix is only based on the geometry of the system and does not depend on the physical properties such as mass and stiffness.

**** Center Of Mass

If the center of mass is chosen as the decoupling frame.
The Jacobian matrix and its inverse are expressed in eqref:eq:detail_control_decoupling_jacobian_CoM_inverse.

\begin{equation}\label{eq:detail_control_decoupling_jacobian_CoM_inverse}
  \bm{J}_{\{M\}} = \begin{bmatrix}
    1 & 0 &  h_a \\
    0 & 1 & -l_a \\
    0 & 1 &  l_a \\
  \end{bmatrix}, \quad \bm{J}_{\{M\}}^{-1} = \begin{bmatrix}
    1 & \frac{h_a}{2 l_a} & \frac{-h_a}{2 l_a} \\
    0 & \frac{1}{2} & \frac{1}{2} \\
    0 & \frac{-1}{2 l_a} & \frac{1}{2 l_a} \\
  \end{bmatrix}
\end{equation}

#+begin_src latex :file detail_control_decoupling_control_jacobian_CoM.pdf
\begin{tikzpicture}
  \node[block] (G) {$\bm{G}_{\{\mathcal{L}\}}$};
  \node[block, left=0.6 of G] (Jt) {$\bm{J}_{\{M\}}^{-\intercal}$};
  \node[block, right=0.6 of G] (Ja) {$\bm{J}_{\{M\}}^{-1}$};

  % Connections and labels
  \draw[<-] (Jt.west) -- ++(-1.4, 0) node[above right]{$\bm{\mathcal{F}}_{\{M\}}$};
  \draw[->] (Jt.east) -- (G.west)  node[above left]{$\bm{\tau}$};
  \draw[->] (G.east) -- (Ja.west)  node[above left]{$\bm{\mathcal{L}}$};
  \draw[->] (Ja.east) -- ++( 1.4, 0)  node[above left]{$\bm{\mathcal{X}}_{\{M\}}$};

  \begin{scope}[on background layer]
    \node[fit={(Jt.south west) (Ja.north east)}, fill=black!10!white, draw, dashed, inner sep=4pt] (Gx) {};
    \node[above] at (Gx.north) {$\bm{G}_{\{M\}}$};
  \end{scope}
\end{tikzpicture}
#+end_src

#+RESULTS:
[[file:figs/detail_control_decoupling_control_jacobian_CoM.png]]
# [[file:figs/detail_control_decoupling_control_jacobian_CoM.png]]

Analytical formula of the plant is eqref:eq:detail_control_decoupling_plant_CoM.

\begin{equation}\label{eq:detail_control_decoupling_plant_CoM}
  \frac{\bm{\mathcal{X}}_{\{M\}}}{\bm{\mathcal{F}}_{\{M\}}}(s) = \bm{G}_{\{M\}}(s) = \left( \bm{M}_{\{M\}} s^2 + \bm{J}_{\{M\}}^{\intercal} \bm{\mathcal{C}} \bm{J}_{\{M\}} s + \bm{J}_{\{M\}}^{\intercal} \bm{\mathcal{K}} \bm{J}_{\{M\}} \right)^{-1}
\end{equation}

At high frequency, converges towards the inverse of the mass matrix, which is a diagonal matrix eqref:eq:detail_control_decoupling_plant_CoM_high_freq.

\begin{equation}\label{eq:detail_control_decoupling_plant_CoM_high_freq}
  \bm{G}_{\{M\}}(j\omega) \xrightarrow[\omega \to \infty]{} -\omega^2 \bm{M}_{\{M\}}^{-1} = -\omega^2 \begin{bmatrix}
    1/m & 0   & 0 \\
    0   & 1/m & 0 \\
    0   & 0   & 1/I
  \end{bmatrix}
\end{equation}

Plant is therefore well decoupled above the suspension mode with the highest frequency.
Such strategy is usually applied on systems with low frequency suspension modes, such that the plant corresponds to decoupled mass lines.

- [ ] Reference to some papers about vibration isolation or ASML?

The coupling at low frequency can easily be understood physically.
When a static (or with frequency lower than the suspension modes) force is applied at the center of mass, rotation is induced by the stiffness of the first actuator, not in line with the force application point.
this is illustrated in Figure ref:fig:detail_control_decoupling_model_test_CoM.

#+begin_src matlab
%% Jacobian Decoupling - Center of Mass
G_CoM = pinv(J_CoM)*G_L*pinv(J_CoM');
G_CoM.InputName  = {'Fx', 'Fy', 'Mz'};
G_CoM.OutputName  = {'Dx', 'Dy', 'Rz'};
#+end_src

#+begin_src matlab :exports none
figure;
hold on;
plot(freqs, abs(squeeze(freqresp(G_CoM(1, 3), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
     'DisplayName', '$D_{x,\{M\}}/M_{z,\{M\}}$');
plot(freqs, abs(squeeze(freqresp(G_CoM(3, 1), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
     'DisplayName', '$R_{z,\{M\}}/F_{x,\{M\}}$');
plot(freqs, abs(squeeze(freqresp(G_CoM(1, 1), freqs, 'Hz'))), 'color', colors(1,:), 'DisplayName', '$D_{x,\{M\}}/F_{x,\{M\}}$');
plot(freqs, abs(squeeze(freqresp(G_CoM(2, 2), freqs, 'Hz'))), 'color', colors(2,:), 'DisplayName', '$D_{y,\{M\}}/F_{y,\{M\}}$');
plot(freqs, abs(squeeze(freqresp(G_CoM(3, 3), freqs, 'Hz'))), 'color', colors(3,:), 'DisplayName', '$R_{z,\{M\}}/M_{z,\{M\}}$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
ylim([1e-10, 1e-3]);
leg = legend('location', 'southwest', 'FontSize', 8);
leg.ItemTokenSize(1) = 18;
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/detail_control_decoupling_jacobian_plant_CoM.pdf', 'width', 'half', 'height', 'normal');
#+end_src

#+name: fig:detail_control_jacobian_decoupling_plant_CoM_results
#+caption: Plant decoupled using the Jacobian matrix expresssed at the center of mass (\subref{fig:detail_control_decoupling_jacobian_plant_CoM}). The physical reason for low frequency coupling is illustrated in (\subref{fig:detail_control_decoupling_model_test_CoM}).
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:detail_control_decoupling_jacobian_plant_CoM}Dynamics at the CoM}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
#+attr_latex: :width 0.95\linewidth
[[file:figs/detail_control_decoupling_jacobian_plant_CoM.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_control_decoupling_model_test_CoM}Static force applied at the CoM}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
#+attr_latex: :scale 1
[[file:figs/detail_control_decoupling_model_test_CoM.png]]
#+end_subfigure
#+end_figure

**** Center Of Stiffness

#+begin_src latex :file detail_control_decoupling_control_jacobian_CoK.pdf
\begin{tikzpicture}
  \node[block] (G) {$\bm{G}_{\{\mathcal{L}\}}$};
  \node[block, left=0.6 of G] (Jt) {$\bm{J}_{\{K\}}^{-\intercal}$};
  \node[block, right=0.6 of G] (Ja) {$\bm{J}_{\{K\}}^{-1}$};

  % Connections and labels
  \draw[<-] (Jt.west) -- ++(-1.4, 0) node[above right]{$\bm{\mathcal{F}}_{\{K\}}$};
  \draw[->] (Jt.east) -- (G.west)  node[above left]{$\bm{\tau}$};
  \draw[->] (G.east) -- (Ja.west)  node[above left]{$\bm{\mathcal{L}}$};
  \draw[->] (Ja.east) -- ++( 1.4, 0)  node[above left]{$\bm{\mathcal{X}}_{\{K\}}$};

  \begin{scope}[on background layer]
    \node[fit={(Jt.south west) (Ja.north east)}, fill=black!10!white, draw, dashed, inner sep=4pt] (Gx) {};
    \node[above] at (Gx.north) {$\bm{G}_{\{K\}}$};
  \end{scope}
\end{tikzpicture}
#+end_src

#+RESULTS:
[[file:figs/detail_control_decoupling_control_jacobian_CoK.png]]
# [[file:figs/detail_control_decoupling_control_jacobian_CoK.png]]

\begin{equation}
  \bm{J}_{\{K\}} = \begin{bmatrix}
    1 & 0 &  0   \\
    0 & 1 & -l_a \\
    0 & 1 &  l_a
  \end{bmatrix}, \quad \bm{J}_{\{K\}}^{-1} = \begin{bmatrix}
    1 & 0                & 0   \\
    0 & \frac{1}{2}      & \frac{1}{2} \\
    0 & \frac{-1}{2 l_a} & \frac{1}{2 l_a}
  \end{bmatrix}
\end{equation}

Frame $\{K\}$ is chosen such that $\bm{J}_{\{K\}}^{\intercal} \bm{\mathcal{K}} \bm{J}_{\{K\}}$ is diagonal.
Typically, it can me made based on physical reasoning as is the case here.

\begin{equation}\label{eq:detail_control_decoupling_plant_CoK}
  \frac{\bm{\mathcal{X}}_{\{K\}}}{\bm{\mathcal{F}}_{\{K\}}}(s) = \bm{G}_{\{K\}}(s) = \left( \bm{J}_{\{K\}}^{\intercal} \bm{J}_{\{M\}}^{-\intercal} \bm{M}_{\{M\}} \bm{J}_{\{M\}}^{-1} \bm{J}_{\{K\}} s^2 + \bm{J}_{\{K\}}^{\intercal} \bm{\mathcal{C}} \bm{J}_{\{K\}} s + \bm{J}_{\{K\}}^{\intercal} \bm{\mathcal{K}} \bm{J}_{\{K\}} \right)^{-1}
\end{equation}

Plant is well decoupled below the suspension mode with the lowest frequency.
This is usually suited for systems which high stiffness.

\begin{equation}
  \bm{G}_{\{K\}}(j\omega) \xrightarrow[\omega \to 0]{} \bm{J}_{\{K\}}^{-1} \bm{\mathcal{K}}^{-1} \bm{J}_{\{K\}}^{-\intercal}
\end{equation}


The physical reason for high frequency coupling is schematically shown in Figure ref:fig:detail_control_decoupling_model_test_CoK.
At high frequency, a force applied on a point which is not aligned with the center of mass.
Therefore, it will induce some rotation around the center of mass.

#+begin_src matlab
%% Jacobian Decoupling - Center of Mass
%  Location of the joints with respect to the center of stiffness
Mb1 = [-l/2; 0];
Mb2 = [-la; -h/2+ha];
Mb3 = [ la; -h/2+ha];

% Jacobian matrix (Center of Stiffness)
J_CoK = [s1', Mb1(1)*s1(2)-Mb1(2)*s1(1);
         s2', Mb2(1)*s2(2)-Mb2(2)*s2(1);
         s3', Mb3(1)*s3(2)-Mb3(2)*s3(1)];

G_CoK = pinv(J_CoK)*G_L*pinv(J_CoK');
G_CoK.InputName   = {'Fx', 'Fy', 'Mz'};
G_CoK.OutputName  = {'Dx', 'Dy', 'Rz'};
#+end_src

#+begin_src matlab :exports none
figure;
hold on;
plot(freqs, abs(squeeze(freqresp(G_CoK(1, 1), freqs, 'Hz'))), 'color', colors(1,:), 'DisplayName', '$D_{x,\{K\}}/F_{x,\{K\}}$');
plot(freqs, abs(squeeze(freqresp(G_CoK(2, 2), freqs, 'Hz'))), 'color', colors(2,:), 'DisplayName', '$D_{y,\{K\}}/F_{y,\{K\}}$');
plot(freqs, abs(squeeze(freqresp(G_CoK(3, 3), freqs, 'Hz'))), 'color', colors(3,:), 'DisplayName', '$R_{z,\{K\}}/M_{z,\{K\}}$');
plot(freqs, abs(squeeze(freqresp(G_CoK(1, 3), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
     'DisplayName', '$D_{x,\{K\}}/M_{z,\{K\}}$');
plot(freqs, abs(squeeze(freqresp(G_CoK(3, 1), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
     'DisplayName', '$R_{z,\{K\}}/F_{x,\{K\}}$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Kagnitude');
ylim([1e-10, 1e-3]);
leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
leg.ItemTokenSize(1) = 18;
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/detail_control_decoupling_jacobian_plant_CoK.pdf', 'width', 'half', 'height', 'normal');
#+end_src

#+name: fig:detail_control_decoupling_jacobian_plant_CoK_results
#+caption: Plant decoupled using the Jacobian matrix expresssed at the center of stiffness (\subref{fig:detail_control_decoupling_jacobian_plant_CoK}). The physical reason for high frequency coupling is illustrated in (\subref{fig:detail_control_decoupling_model_test_CoK}).
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:detail_control_decoupling_jacobian_plant_CoK}Dynamics at the CoK}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
#+attr_latex: :width 0.95\linewidth
[[file:figs/detail_control_decoupling_jacobian_plant_CoK.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_control_decoupling_model_test_CoK}High frequency force applied at the CoK}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
#+attr_latex: :scale 1
[[file:figs/detail_control_decoupling_model_test_CoK.png]]
#+end_subfigure
#+end_figure

** Modal Decoupling
<<ssec:detail_control_decoupling_modal>>
**** Theory                                                        :ignore:

- A mechanical system consists of several modes:
  - Modal decomposition [[cite:&rankers98_machin]]
  #+begin_quote
The physical interpretation of the above two equations is that any motion of the system can be regarded as a combination of the contribution of the various modes.
  #+end_quote
  - Mode superposition [[cite:&preumont94_random_vibrat_spect_analy;&preumont18_vibrat_contr_activ_struc_fourt_edition, chapt. 2]]
- The idea is to control the system in the "modal space"
  [[cite:&heertjes05_activ_vibrat_isolat_metrol_frames]]
  IFF in modal space [[cite:&holterman05_activ_dampin_based_decoup_colloc_contr]] very interesting paper
  [[cite:&pu11_six_degree_of_freed_activ]]

\begin{equation}\label{eq:detail_control_decoupling_equation_motion_CoM}
  \bm{M}_{\{M\}} \ddot{\bm{\mathcal{X}}}_{\{M\}}(t) + \bm{C}_{\{M\}} \dot{\bm{\mathcal{X}}}_{\{M\}}(t) + \bm{K}_{\{M\}} \bm{\mathcal{X}}_{\{M\}}(t) = \bm{J}_{\{M\}}^{\intercal} \bm{\tau}(t)
\end{equation}

Let's make a change of variables:
\begin{equation}\label{eq:detail_control_decoupling_modal_coordinates}
  \bm{\mathcal{X}}_{\{M\}} = \bm{\Phi} \bm{\mathcal{X}}_{m}
\end{equation}
with:
- $\bm{\mathcal{X}}_{m}$ the modal amplitudes
- $\bm{\Phi}$ a matrix whose columns are the modes shapes of the system which can be computed from $\bm{M}_{\{M\}}$ and $\bm{K}_{\{M\}}$.

By pre-multiplying the equation of motion eqref:eq:detail_control_decoupling_equation_motion_CoM by $\bm{\Phi}^{\intercal}$ and using the change of variable eqref:eq:detail_control_decoupling_modal_coordinates, a new set of equation of motion are obtained

\begin{equation}\label{eq:detail_control_decoupling_equation_modal_coordinates}
  \underbrace{\bm{\Phi}^{\intercal} \bm{M} \bm{\Phi}}_{\bm{M}_m} \bm{\ddot{\mathcal{X}}}_m(t) + \underbrace{\bm{\Phi}^{\intercal} \bm{C} \bm{\Phi}}_{\bm{C}_m} \bm{\dot{\mathcal{X}}}_m(t)  + \underbrace{\bm{\Phi}^{\intercal} \bm{K} \bm{\Phi}}_{\bm{K}_m} \bm{\mathcal{X}}_m(t) = \underbrace{\bm{\Phi}^{\intercal} \bm{J}^{\intercal} \bm{\tau}(t)}_{\bm{\tau}_m(t)}
\end{equation}

- $\bm{\tau}_m$ is the modal input
- $\bm{M}_m$, $\bm{C}_m$ and $\bm{K}_m$ are the modal mass, damping and stiffness matrices

Orthogonality of normal modes gives that the "the modal
vectors uncouple the equations of motion making each dynamic equation independent of all the others" [[cite:&lang17_under]].
The modal matrices are diagonal.

In order to implement such modal decoupling from the decentralized plant, architecture shown in Figure ref:fig:detail_control_decoupling_modal can be used.
The dynamics from modal inputs $\bm{\tau}_m$ to modal amplitudes $\bm{\mathcal{X}}_m$ is fully decoupled.

#+begin_src latex :file detail_control_decoupling_modal.pdf
\begin{tikzpicture}
  \node[block] (G) {$\bm{G}_{\{\mathcal{L}\}}$};
  \node[block, left=0.6 of G] (Jt) {$\bm{J}_{\{M\}}^{-\intercal}$};
  \node[block, left=1.2 of Jt] (Bm) {$\bm{\Phi}^{-\intercal}$};
  \node[block, right=0.6 of G] (J) {$\bm{J}_{\{M\}}^{-1}$};
  \node[block, right=1.2 of J] (Cm) {$\bm{\Phi}^{-1}$};

  % Connections and labels
  \draw[<-] (Bm.west) -- ++(-1.0, 0) node[above right]{$\bm{\tau}_m$};
  \draw[->] (Bm.east) -- (Jt.west)  node[above left]{$\bm{\mathcal{F}}_{\{M\}}$};
  \draw[->] (Jt.east) -- (G.west)  node[above left]{$\bm{\tau}$};
  \draw[->] (G.east) -- (J.west)  node[above left]{$\bm{\mathcal{L}}$};
  \draw[->] (J.east) -- (Cm.west)  node[above left]{$\bm{\mathcal{X}}_{\{M\}}$};
  \draw[->] (Cm.east) -- ++( 1.0, 0)  node[above left]{$\bm{\mathcal{X}}_m$};

  \begin{scope}[on background layer]
    \node[fit={(Bm.south west) (Cm.north east)}, fill=black!10!white, draw, dashed, inner sep=4pt] (Gm) {};
    \node[above] at (Gm.north) {$\bm{G}_m$};
  \end{scope}
\end{tikzpicture}
#+end_src

#+name: fig:detail_control_decoupling_modal
#+caption: Modal Decoupling Architecture
#+RESULTS:
[[file:figs/detail_control_decoupling_modal.png]]

Modal decoupling requires to have the equations of motion of the system.
From the equations of motion (and more precisely the mass and stiffness matrices), the mode shapes $\Phi$ are computed.

Then, the system can be decoupled in the modal space.
The obtained system on the diagonal are second order resonant systems which can be easily controlled.

Using this decoupling strategy, it is possible to control each mode individually.

- [ ] Do we need to measure all the states?
  I think so
- [ ] Say that the eigen vectors are unitary
  Are they orthogonal?
- [ ] Say that the obtained plant are second order systems

**** Example

From the mass matrix $\bm{M}_{\{M\}}$ and stiffness matrix $\bm{K}_{\{M\}}$ expressed at the center of mass, the eigenvectors of $\bm{M}_{\{M\}}^{-1}\bm{K}_{\{M\}}$ are computed.

\begin{equation}
  \bm{M}_{\{M\}} = \begin{bmatrix}
    m & 0 & 0 \\
    0 & m & 0 \\
    0 & 0 & I
  \end{bmatrix}, \quad
  \bm{K}_{\{M\}} = \begin{bmatrix}
    k & 0 & 0 \\
    0 & k & 0 \\
    0 & 0 & k
  \end{bmatrix}
\end{equation}

Obtained

\begin{equation}
\bm{\Phi} = \begin{bmatrix}
\frac{I - h_a^2 m - 2 l_a^2 m - \alpha}{2 h_a m} & 0 & \frac{I - h_a^2 m - 2 l_a^2 m + \alpha}{2 h_a m} \\
0                                                & 1 & 0                                                \\
1                                                & 0 & 1
\end{bmatrix},\ \alpha = \sqrt{\left( I + m (h_a^2 - 2 l_a^2) \right)^2 + 8 m^2 h_a^2 l_a^2}
\end{equation}

It may be very difficult to obtain eigenvectors analytically, so typically these can be computed numerically.

For the present test system, obtained eigen vectors are

Eigenvectors are arranged for increasing eigenvalues (i.e. resonance frequencies).

\begin{equation}
  \bm{\phi} = \begin{bmatrix}
    -0.905 & 0 & -0.058 \\
     0     & 1 &  0     \\
     0.424 & 0 & -0.998
  \end{bmatrix}, \quad
  \bm{\phi}^{-1} = \begin{bmatrix}
    -1.075 & 0 &  0.063 \\
     0     & 1 &  0     \\
    -0.457 & 0 & -0.975
  \end{bmatrix}
\end{equation}

- [ ] Formula for the plant transfer function

#+begin_src matlab
%% Modal decoupling
% Compute the eigen vectors
[phi, wi] = eig(M\K);
% Sort the eigen vectors by increasing associated frequency
[~, i] = sort(diag(wi));
phi = phi(:, i);

% Plant in the modal space
Gm = inv(phi)*inv(J_CoM)*G_L*inv(J_CoM')*inv(phi');
#+end_src

#+begin_src matlab :exports none
%% Modal decoupled plant
figure;
hold on;
plot(freqs, abs(squeeze(freqresp(Gm(1,1), freqs, 'Hz'))), 'color', colors(1,:), 'DisplayName', '$\mathcal{X}_{m,1}/\tau_{m,1}$');
plot(freqs, abs(squeeze(freqresp(Gm(2,2), freqs, 'Hz'))), 'color', colors(2,:), 'DisplayName', '$\mathcal{X}_{m,2}/\tau_{m,2}$');
plot(freqs, abs(squeeze(freqresp(Gm(3,3), freqs, 'Hz'))), 'color', colors(3,:), 'DisplayName', '$\mathcal{X}_{m,3}/\tau_{m,3}$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
ylim([1e-8, 1e-4]);
leg = legend('location', 'northeast', 'FontSize', 8);
leg.ItemTokenSize(1) = 18;
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/detail_control_decoupling_modal_plant.pdf', 'width', 'half', 'height', 'normal');
#+end_src

#+name: fig:detail_control_decoupling_modal_plant_modes
#+caption: Plant using modal decoupling consists of second order plants (\subref{fig:detail_control_decoupling_modal_plant}) which can be used to control separately different modes (\subref{fig:detail_control_decoupling_model_test_modal})
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:detail_control_decoupling_modal_plant}Decoupled plant in modal space}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
#+attr_latex: :width 0.95\linewidth
[[file:figs/detail_control_decoupling_modal_plant.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_control_decoupling_model_test_modal}Individually controlled modes}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
#+attr_latex: :width 0.95\linewidth
[[file:figs/detail_control_decoupling_model_test_modal.png]]
#+end_subfigure
#+end_figure

** SVD Decoupling
<<ssec:detail_control_decoupling_svd>>
**** Singular Value Decomposition

Singular Value Decomposition (SVD)
- Introduction to SVD [[cite:&brunton22_data, chapt. 1]]
- Singular value is used a lot for multivariable control [[cite:&skogestad07_multiv_feedb_contr]].
  Used to study directions in multivariable systems.

# Should I consider only real matrices?

The SVD is a unique matrix decomposition that exists for every complex matrix $\bm{X} \in \mathbb{C}^{n \times m}$.

\begin{equation}\label{eq:detail_control_svd}
  \bm{X} = \bm{U} \bm{\Sigma} \bm{V}^H
\end{equation}

where $\bm{U} \in \mathbb{C}^{n \times n}$ and $\bm{V} \in \mathbb{C}^{m \times m}$ are unitary matrices with orthonormal columns, and $\bm{\Sigma} \in \mathbb{R}^{n \times n}$ is a diagonal matrix with real, non-negative entries on the diagonal.

If the matrix $\bm{X}$ is a real matrix, the obtained $\bm{U}$ and $\bm{V}$ matrices are real and can be used for decoupling purposes.

The idea to use Singular Value Decomposition as a way to decouple a plant is not new
- [ ] Quick review of SVD controllers
  [[cite:&skogestad07_multiv_feedb_contr, chapt. 3.5.4]]

**** Decoupling using the SVD

*Procedure*:
Identify the dynamics of the system from inputs to outputs (can be obtained experimentally)
Frequency Response Function, which is a complex matrix obtained for several frequency points $\bm{G}(\omega_i)$.


Choose a frequency where we want to decouple the system (usually, the crossover frequency $\omega_c$ is a good choice)

As /real/ V and U matrices need to be obtained, a real approximation of the complex measured response needs to be computed.
Compute a real approximation of the system's response at that frequency.
[[cite:&kouvaritakis79_theor_pract_charac_locus_desig_method]]: real matrix that preserves the most orthogonality in directions with the input complex matrix

Then, a real matrix $\tilde{\bm{G}}(\omega_c)$ is obtained, and the SVD is performed on this real matrix.
Unitary $\bm{U}$ and $\bm{V}$ matrices are then obtained such that $\bm{V}^{-\intercal} \tilde{\bm{G}}(\omega_c) \bm{U}^{-1}$ is diagonal.

Use the singular input and output matrices to decouple the system as shown in Figure ref:fig:detail_control_decoupling_svd

\begin{equation}
  G_{\text{SVD}}(s) = \bm{U}^{-1} \bm{G}_{\{\mathcal{L}\}}(s) \bm{V}^{-\intercal}
\end{equation}

#+begin_src latex :file detail_control_decoupling_svd.pdf
\begin{tikzpicture}
  \node[block] (G) {$\bm{G}_{\{\mathcal{L}\}}$};
  \node[block, left=0.6 of G.west] (V) {$\bm{V}^{-\intercal}$};
  \node[block, right=0.6 of G.east] (U) {$\bm{U}^{-1}$};

  % Connections and labels
  \draw[<-] (V.west) -- ++(-0.8, 0) node[above right]{$\bm{u}$};
  \draw[->] (V.east) -- (G.west) node[above left]{$\bm{\tau}$};
  \draw[->] (G.east) -- (U.west) node[above left]{$\bm{\mathcal{L}}$};
  \draw[->] (U.east) -- ++( 0.8, 0) node[above left]{$\bm{y}$};

  \begin{scope}[on background layer]
    \node[fit={(V.south west) (G.north-|U.east)}, fill=black!10!white, draw, dashed, inner sep=4pt] (Gsvd) {};
    \node[above] at (Gsvd.north) {$\bm{G}_{\text{SVD}}$};
  \end{scope}
\end{tikzpicture}
#+end_src

#+name: fig:detail_control_decoupling_svd
#+caption: Decoupled plant $\bm{G}_{\text{SVD}}$ using the Singular Value Decomposition
#+RESULTS:
[[file:figs/detail_control_decoupling_svd.png]]

In order to apply the Singular Value Decomposition, we need to have the Frequency Response Function of the system, at least near the frequency where we wish to decouple the system.
The FRF can be experimentally obtained or based from a model.

This method ensure good decoupling near the chosen frequency, but no guaranteed decoupling away from this frequency.

Also, it depends on how good the real approximation of the FRF is, therefore it might be less good for plants with high damping.

This method is quite general and can be applied to any type of system.
The inputs and outputs are ordered from higher gain to lower gain at the chosen frequency.

- [ ] Do we loose any physical meaning of the obtained inputs and outputs?
- [ ] Can we take advantage of the fact that U and V are unitary?

**** Example


\begin{equation}
\begin{align}
  & \bm{G}_{\{\mathcal{L}\}}(\omega_c) = 10^{-9} \begin{bmatrix}
    -99 - j 2.6 &  74  + j 4.2 & -74  - j 4.2 \\
     74 + j 4.2 & -247 - j 9.7 &  102 + j 7.0 \\
    -74 - j 4.2 &  102 + j 7.0 & -247 - j 9.7
  \end{bmatrix} \\
  & \xrightarrow[\text{approximation}]{\text{real}} \tilde{\bm{G}}_{\{\mathcal{L}\}(\omega_c)} = 10^{-9} \begin{bmatrix}
    -99 &  74  & -74  \\
     74 & -247 &  102 \\
    -74 &  102 & -247
  \end{bmatrix} \\
  & \xrightarrow[\text{SVD}]{\phantom{\text{approximation}}} \bm{U} = \begin{bmatrix}
    0.34 & 0    &  0.94 \\
   -0.66 & 0.71 &  0.24 \\
    0.66 & 0.71 & -0.24
  \end{bmatrix}, \ \bm{V} = \begin{bmatrix}
    -0.34 &  0    & -0.94 \\
     0.66 & -0.71 & -0.24 \\
    -0.66 & -0.71 &  0.24
  \end{bmatrix}
\end{align}
\end{equation}

Once the $\bm{U}$ and $\bm{V}$ matrices are obtained, the decoupled plant can be computed using eqref:eq:detail_control_decoupling_plant_svd.

\begin{equation}\label{eq:detail_control_decoupling_plant_svd}
  \bm{G}_{\text{SVD}}(s) = \bm{U}^{-1} \bm{G}_{\{\mathcal{L}\}}(s) \bm{V}^{-\intercal}
\end{equation}

The obtained plant shown in Figure ref:fig:detail_control_decoupling_svd_plant is very well decoupled. and not only around $\omega_c$.
On top of that, the diagonal terms are second order plants.

#+begin_src matlab
%% SVD Decoupling
wc = 2*pi*100; % Decoupling frequency [rad/s]
% System's response at the decoupling frequency
H1 = evalfr(G_L, j*wc);

% Real approximation of G(j.wc)
D = pinv(real(H1'*H1));
H1 = pinv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));

[U,S,V] = svd(H1);

Gsvd = inv(U)*G_L*inv(V');
#+end_src

#+begin_src matlab :exports none
figure;
hold on;
for i_in = 1:3
    for i_out = [i_in+1:3]
        plot(freqs, abs(squeeze(freqresp(Gsvd(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
             'HandleVisibility', 'off');
    end
end
plot(freqs, abs(squeeze(freqresp(Gsvd(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
     'DisplayName', '$G_{SVD}(i,j)\ i \neq j$');
set(gca,'ColorOrderIndex',1)
for i_in_out = 1:3
    plot(freqs, abs(squeeze(freqresp(Gsvd(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_{SVD}(%d,%d)$', i_in_out, i_in_out));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
ylim([1e-10, 2e-4]);
leg = legend('location', 'northeast', 'FontSize', 8);
leg.ItemTokenSize(1) = 18;
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/detail_control_decoupling_svd_plant.pdf', 'width', 'wide', 'height', 'normal');
#+end_src

#+name: fig:detail_control_decoupling_svd_plant
#+caption: Svd plant $G_m(s)$
#+RESULTS:
[[file:figs/detail_control_decoupling_svd_plant.png]]



- [ ] Do we have something special when applying SVD to a collocated MIMO system?
  As shown in Figure ref:fig:detail_control_decoupling_coupled_plant_bode, the plant is symmetrical.
  Paper by Skogestad mention that.
  "symmetric circular plants" [[cite:&hovd97_svd_contr_contr]]




A second system, identical to the first in terms of dynamics.
Just the sensor are changed.
Instead of having relative motion sensors in the frame of the struts, three relative motion sensors are used as shown in Figure ref:fig:detail_control_decoupling_model_test_alt.
Using Jacobian matrices, it is possible to compute the relative motion of each struts.
So theoretically, it should be possible to control both systems the same way.

However, when applying the same SVD decoupling, plant of Figure ref:fig:detail_control_decoupling_svd_alt_plant is obtained.
It has much more coupling.
It is interesting to note that the coupling have local minimum near the chosen decoupling frequency.
This is very logical as the decoupling matrices were computed from the plant response at that particular frequency.

#+begin_src matlab
%% Simscape model with relative motion sensor at alternative positions
mdl = 'detail_control_decoupling_test_model';
open(mdl)

deq = 0.2; % Length of the actuators [m]

% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/F1'], 1, 'openinput');  io_i = io_i + 1;
io(io_i) = linio([mdl, '/F2'], 1, 'openinput');  io_i = io_i + 1;
io(io_i) = linio([mdl, '/F3'], 1, 'openinput');  io_i = io_i + 1;
io(io_i) = linio([mdl, '/Payload'], 1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Payload'], 2, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Payload'], 3, 'openoutput'); io_i = io_i + 1;

G_L_alt = linearize(mdl, io);
G_L_alt.InputName  = {'F1', 'F2', 'F3'};
G_L_alt.OutputName = {'d1', 'd2', 'd32'};

% SVD Decoupling with the new plant
wc = 2*pi*100; % Decoupling frequency [rad/s]
% System's response at the decoupling frequency
H1 = evalfr(G_L_alt, j*wc);

% Real approximation of G(j.wc)
D = pinv(real(H1'*H1));
H1 = pinv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));

[U,S,V] = svd(H1);

Gsvd_alt = inv(U)*G_L_alt*inv(V');
#+end_src

#+begin_src matlab :exports none :results none
%% Obtained plant after SVD decoupling - Relative motion sensors are not collocated with the actuators
figure;
hold on;
for i_in = 1:3
    for i_out = [i_in+1:3]
        plot(freqs, abs(squeeze(freqresp(Gsvd_alt(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
             'HandleVisibility', 'off');
    end
end
plot(freqs, abs(squeeze(freqresp(Gsvd_alt(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
     'DisplayName', '$G_{SVD}(i,j)\ i \neq j$');
set(gca,'ColorOrderIndex',1)
for i_in_out = 1:3
    plot(freqs, abs(squeeze(freqresp(Gsvd_alt(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_{SVD}(%d,%d)$', i_in_out, i_in_out));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
ylim([5e-11, 7e-5]);
leg = legend('location', 'southwest', 'FontSize', 8);
leg.ItemTokenSize(1) = 18;
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/detail_control_decoupling_svd_alt_plant.pdf', 'width', 'half', 'height', 'normal');
#+end_src

#+name: fig:detail_control_svd_decoupling_not_symmetrical
#+caption: Application of SVD decoupling on a system schematically shown in (\subref{fig:detail_control_decoupling_model_test_alt}). The obtained decoupled plant is shown in (\subref{fig:detail_control_decoupling_svd_alt_plant}).
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:detail_control_decoupling_model_test_alt}Alternative location of sensors}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
#+attr_latex: :scale 1
[[file:figs/detail_control_decoupling_model_test_alt.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_control_decoupling_svd_alt_plant}Obtained decoupled plant}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
#+attr_latex: :width 0.95\linewidth
[[file:figs/detail_control_decoupling_svd_alt_plant.png]]
#+end_subfigure
#+end_figure

** Comparison of decoupling strategies
<<ssec:detail_control_decoupling_comp>>

The three proposed methods may seem very similar as each of them consists of pre-multiplying and post-multiplying the plant with constant matrices.
However, the three methods also differs by a number of points which are summarized in Table ref:tab:detail_control_decoupling_strategies_comp.

However, each method is quite different in terms of approach, and have different pros and cons.

- Comparison of the three proposed methods
- Different "approach" for the three methods:
  - Jacobian is based on geometry
  - Modal decoupling is based on dynamical equations
  - Singular Value Decoupling is based on measured frequency response function
- Depending on the decoupling method, the physical interpretation of inputs and outputs:
  - With Jacobian decoupling, the inputs and outputs can be easily interpreted physically.
    Inputs correspond to force/torques applied on a particular frames
    Outputs corresponds to translation and rotations expressed on a particular frame
  - With modal decoupling, inputs are arranged to excite individual modes.
    By doing a modal analysis (using a FEA for instance) it can be understood how actuator forces are combined to individually excite the different modes.
    Similarly, the outputs are combined to measure the different modes separately.
  - For singular value decomposition, inputs (resp. outputs) are special directions that are ordered from maximum to minimum controllability (resp. observability), at the chosen frequency.
    For plants such as parallel manipulators, it is difficult to have a physical interpretations of the decoupled plants inputs and outputs.
    - [ ] It is really linked to controllability? (add reference about that)
- Decoupling quality:
  - Jacobian: depending on the choice of frame, the plant may be well decoupled at low frequency (Center of Stiffness) or at high frequency (Center of Mass).
    If the system is designed to have both the CoK and the CoM at the same point, the use of Jacobian matrices may lead to excellent decoupling.
  - Modal: good decoupling is obtained for all frequencies.
    However, this is based on a model of the plant, and differences between the model and the physical implementation may lead to large off-diagonal elements.
    Diagonal elements are expected to be simple 2nd order low pass filters, which are easy to control.
  - SVD: as the decoupling matrices can be computed based on measured data, no model is required.
    Decoupling is expected to be good near the frequency chosen for computing the decoupling matrices, but may depend on how good the real approximation of the plant is for that particular frequency.
    Whether the decoupling quality can be guaranteed away from the chosen frequency is unknown.

- "Frame" of the controllers: important to be able to tuned the controllers linked to performance metrics

# Maybe not necessary
There are other aspects that were not treated here such as:
- how to integrate feedforward path and reference signals

#+name: tab:detail_control_decoupling_strategies_comp
#+caption: Comparison of decoupling strategies
#+attr_latex: :environment tabularx :width \linewidth :align lXXX
#+attr_latex: :center t :booktabs t :font \scriptsize
|                       | *Jacobian*                                                                             | *Modal*                                                               | *SVD*                                                            |
|-----------------------+----------------------------------------------------------------------------------------+-----------------------------------------------------------------------+------------------------------------------------------------------|
| *Philosophy*          | Topology Driven                                                                        | Physics Driven                                                        | Data Driven                                                      |
|-----------------------+----------------------------------------------------------------------------------------+-----------------------------------------------------------------------+------------------------------------------------------------------|
| *Requirements*        | Known geometry                                                                         | Known equations of motion                                             | Identified FRF                                                   |
|-----------------------+----------------------------------------------------------------------------------------+-----------------------------------------------------------------------+------------------------------------------------------------------|
| *Decoupling Matrices* | Decoupling using $\bm{J}_{\{O\}}$ obtained from geometry                               | Decoupling using $\bm{\Phi}$ obtained from modal decomposition        | Decoupling using $\bm{U}$ and $\bm{V}$ obtained from SVD         |
|-----------------------+----------------------------------------------------------------------------------------+-----------------------------------------------------------------------+------------------------------------------------------------------|
| *Decoupled Plant*     | $\bm{G}_{\{O\}}(s) = \bm{J}_{\{O\}}^{-1} \bm{G}_{\mathcal{L}}(s) \bm{J}_{\{O\}}^{-\intercal}$ | $\bm{G}_m(s) = \bm{\Phi}^{-1} \bm{G}_{\mathcal{X}}(s) \bm{\Phi}^{-\intercal}$ | $\bm{G}_{\text{SVD}}(s) = \bm{U}^{-1} \bm{G}(s) \bm{V}^{-\intercal}$ |
|-----------------------+----------------------------------------------------------------------------------------+-----------------------------------------------------------------------+------------------------------------------------------------------|
| *Controller*          | $\bm{K}_{\{O\}}(s) = \bm{J}_{\{O\}}^{-\intercal} \bm{K}_{d}(s) \bm{J}_{\{O\}}^{-1}$   | $\bm{K}_m(s) = \bm{\Phi}^{-\intercal} \bm{K}_{d}(s) \bm{\Phi}^{-1}$  | $\bm{K}_{\text{SVD}}(s) = \bm{V}^{-\intercal} \bm{K}_{d}(s) \bm{U}^{-1}$ |
|-----------------------+----------------------------------------------------------------------------------------+-----------------------------------------------------------------------+------------------------------------------------------------------|
| *Interpretation*      | Forces/Torques to Displacement/Rotation in chosen frame                                | Inputs to excite individual modes                                     | Directions of max to min controllability/observability           |
|                       |                                                                                        | Output to sense individual modes                                      |                                                                  |
|-----------------------+----------------------------------------------------------------------------------------+-----------------------------------------------------------------------+------------------------------------------------------------------|
| *Properties*          | Decoupling at low or high frequency depending on the chosen frame                      | Good decoupling at all frequencies                                    | Good decoupling near the chosen frequency                        |
|-----------------------+----------------------------------------------------------------------------------------+-----------------------------------------------------------------------+------------------------------------------------------------------|
| *Pros*                | Physical inputs / outputs                                                              | Target specific modes                                                 | Good Decoupling near the crossover                               |
|                       | Good decoupling at High frequency (diagonal mass matrix if Jacobian taken at the CoM)  | 2nd order diagonal plant                                              | Very General                                                     |
|                       | Good decoupling at Low frequency (if Jacobian taken at specific point)                 |                                                                       |                                                                  |
|                       | Easy integration of meaningful reference inputs                                        |                                                                       |                                                                  |
|                       |                                                                                        |                                                                       |                                                                  |
|-----------------------+----------------------------------------------------------------------------------------+-----------------------------------------------------------------------+------------------------------------------------------------------|
| *Cons*                | Coupling between force/rotation may be high at low frequency (non diagonal terms in K) | Need analytical equations                                             | Loose the physical meaning of inputs /outputs                    |
|                       | Limited to parallel mechanisms (?)                                                     |                                                                       | Decoupling depends on the real approximation validity            |
|                       | If good decoupling at all frequencies => requires specific mechanical architecture     |                                                                       | Diagonal plants may not be easy to control                       |
|-----------------------+----------------------------------------------------------------------------------------+-----------------------------------------------------------------------+------------------------------------------------------------------|
| *Applicability*       | Parallel Mechanisms                                                                    | Systems whose dynamics that can be expressed with M and K matrices    | Very general                                                     |
|                       | Only small motion for the Jacobian matrix to stay constant                             |                                                                       | Need FRF data (either experimentally or analytically)            |

Conclusion about NASS:
- Prefer to use Jacobian decoupling as we get more physical interpretation
- Also, it is possible to take into account different specifications in the different DoF as the control is in a "frame" which corresponds to the specifications.
  For active damping however, it may be reasonable to work in the modal space as different damping may be applied to different modes [[cite:&holterman05_activ_dampin_based_decoup_colloc_contr]].

* Closed-Loop Shaping using Complementary Filters
:PROPERTIES:
:HEADER-ARGS:matlab+: :tangle matlab/detail_control_3_close_loop_shaping.m
:END:
<<sec:detail_control_cf>>

# file:~/Cloud/research/papers/published/verma19_virtu_senso_fusio_high_preci_contr/elsarticle-template-harv.pdf
# file:~/Cloud/research/papers/dehaeze20_virtu_senso_fusio/index.org

** Introduction                                                      :ignore:

Once the system is properly decoupled using one of the approaches described in Section ref:sec:detail_control_decoupling, a diagonal controller can be tuned.
This consists in tuning several SISO controllers.
There are several ways to design a controller to obtain a given performance while ensuring good robustness properties.

Performances of a feedback system (such as response time, disturbance rejection, ) depends on the obtained closed-loop transfer functions.
For instance sensitivity, transmissibility, etc... Gang of Four.
The specifications can usually be expressed in terms of the shape of these closed-loop transfer functions [[cite:&skogestad07_multiv_feedb_contr, chapt. 3]].

In some cases, "fixed" controller structures are used, with as PI and PID controllers.
In such case the controller coefficients are manually tuned to obtain acceptable performance and robustness.
In many cases, PID+LPF can already lead to performances close to optimal, depending on the plant.


Decoupled Open-Loop Shaping:
- Explain procedure when applying open-loop shaping [[cite:&schmidt20_desig_high_perfor_mechat_third_revis_edition, chapt. 4.4.7]]
  [[cite:&steinbuch16_model_based]]
  #+begin_quote
  The key idea of loop-shaping is the modification of the controller such that the open-
loop is made according to specifications. The reason this works so well, is that the controller
enters linearly into the open-loop transfer l(j!) = g(j!)k(j!), so that it is fast and easy
to reason what is to be changed in the controller. However, in practice all specifications are
of course given in terms of the final system performance, i.e. as closed-loop specifications.
So we should convert the closed loop specs into specs on the open-loop.
  #+end_quote
- The controller is usually manually tuned using a series of Integrators, Leads, Lags, Notches, low pass filters
- There are lots of tools to check stability, robustness margins and performances
- Open-Loop shaping is very popular as the open-loop gain depends linearly on the controller.
  So the open-loop transfer function can easily be shaped by modifying the controller response.
- Different techniques for open loop shaping (choice of optimal open-loop gain shape) [[cite:&lurie02_system_archit_trades_using_bode]]
- But this is open-loop shaping, and it does not directly work on the closed loop transfer functions
- The huge advantage of this technique, is that one can tune the controllers based on the measured FRF of the system.
  No plant model is required.
- This is what was done during the conceptual phase after the plan was decoupled in the frame of the struts.


Model based control:
- Review of model based design (LQG, H-Infinity) applied to Stewart platform [[file:~/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/bibliography.org::*Multivariable Control][Multivariable Control]]
  $\hinf$ loop-shaping [[cite:&skogestad07_multiv_feedb_contr]].
- Difficulty to specify robustness to change of payload mass
- Requires high level of expertise.
  Far from standard in industry.
  Application to Stewart platforms is not demonstrated (or performance increased compared to decoupled control and manual loop shaping).


In this section, an alternative controller synthesis scheme is proposed in which complementary filters are used for directly shaping the closed-loop transfer functions.
In this paper, we propose a new controller synthesis method
- based on the use of complementary high pass and low pass filters
- inverse based control
- direct translation of requirements such as disturbance rejection and robustness to plant uncertainty

** 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

#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src

#+begin_src matlab :tangle no :noweb yes
<<m-init-path>>
#+end_src

#+begin_src matlab :eval no :noweb yes
<<m-init-path-tangle>>
#+end_src

#+begin_src matlab :noweb yes
<<m-init-other>>

%% Initialize Frequency Vector
freqs = logspace(-1, 3, 1000);
#+end_src

** Control Architecture
 <<ssec:detail_control_cf_control_arch>>
**** Virtual Sensor Fusion

Let's consider the control architecture represented in Figure ref:fig:detail_control_cf_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 ($H_L(s) + H_H(s) = 1$).
The signals are the reference signal $r$, the output perturbation $d_y$, the measurement noise $n$ and the control input $u$.

#+begin_src latex :file detail_control_cf_arch.pdf
\tikzset{block/.default={0.8cm}{0.6cm}}
\tikzset{addb/.append style={scale=0.7}}
\tikzset{node distance=0.6}
\def\cdist{0.7}

\begin{tikzpicture}
  \node[addb={+}{}{}{}{-}] (addfb) at (0, 0){};
  \node[block, right=of addfb] (K){$k$};
  \node[block, right=2.5 of K] (G){$G^\prime$};
  \node[addb={+}{}{}{}{}, right=of G] (adddy){};
  \coordinate[] (KG) at ($(K.east)+(0.3, 0)$);
  \node[block, below=of KG] (Gm){$G$};
  \node[block, below=0.4 of Gm] (Hh){$H_H$};
  \node[addb={+}{}{}{}{}, below=0.4 of Hh] (addcf){};
  \node[block, right= of addcf] (Hl) {$H_L$};
  \node[addb={+}{}{}{}{}, right=2.5 of Hl] (addn) {};


  \draw[->] (addfb.east) -- (K.west) node[above left]{};
  \draw[->] (K.east) -- (G.west) node[above left]{$u$};
  \draw[->] (KG) node[branch]{} -- (Gm.north);
  \draw[->] (Gm.south) -- (Hh.north);
  \draw[->] (Hh.south) -- (addcf.north) node[above left]{};
  \draw[->] (Hl.west) -- (addcf.east);
  \draw[->] (addcf.west) -| (addfb.south) node[below right]{};
  \draw[->] (G.east) -- (adddy.west);
  \draw[<-] (addn.east) -- ++(\cdist, 0) coordinate[](endpos) node[above left]{$n$};
  \draw[->] (adddy.east) -- (G-|endpos) node[above left]{$y$};
  \draw[->] (adddy-|addn) node[branch]{} -- (addn.north);
  \draw[<-] (addfb.west) -- ++(-\cdist, 0) node[above right](r){$r$};
  \draw[->] (addn.west) -- (Hl.east) node[above right]{$y_m$};
  \draw[<-] (adddy.north) -- ++(0, \cdist) node[below right]{$d_y$};

  \begin{scope}[on background layer]
    \node[fit={(Hl.south east) (r.north west)}, inner sep=4pt, draw, fill=black!20!white, dashed, label={RT controller}] (Kfb) {};
  \end{scope}
\end{tikzpicture}
#+end_src

#+name: fig:detail_control_cf_arch
#+caption: Sensor Fusion Architecture
#+RESULTS:
[[file:figs/detail_control_cf_arch.png]]

The dynamics of the closed-loop system is described by eqref:eq:detail_control_cf_sf_cl_tf with $L = k(G H_H + G^\prime H_L)$.

\begin{subequations}\label{eq:detail_control_cf_sf_cl_tf}
  \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}
\end{subequations}

The idea of using such architecture comes from sensor fusion [[cite:&collette14_vibrat;&collette15_sensor_fusion_method_high_perfor]] where two sensors are used.
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.
This lead to the idea of virtual sensor fusion presented in [[cite:&verma20_virtual_sensor_fusion_high_precis_contr]].

One may think that the control architecture shown in Figure ref:fig:detail_control_cf_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 Figure ref:fig:detail_control_cf_arch_eq.

#+begin_src latex :file detail_control_cf_arch_eq.pdf
\tikzset{block/.default={0.8cm}{0.6cm}}
\tikzset{addb/.append style={scale=0.7}}
\tikzset{node distance=0.6}
\def\cdist{0.7}

\begin{tikzpicture}
  \node[addb={+}{}{}{}{-}] (addfb) at (0, 0){};
  \node[addb={+}{}{}{}{-}, right=of addfb] (addK){};
  \node[block, right=of addK] (K){$k$};
  \node[block, right=1.8 of K] (G){$G^\prime$};
  \node[addb={+}{}{}{}{}, right=of G] (adddy){};
  \node[block, below right=0.5 and -0.2 of K] (Gm){$G$};
  \node[block, below left =0.5 and -0.2 of K] (Hh){$H_H$};
  \node[block, below=1.5 of K] (Hl) {$H_L$};
  \node[addb={+}{}{}{}{}, right=3.8 of Hl] (addn) {};

  \draw[->] (addfb.east) -- (addK.west);
  \draw[->] (addK.east) -- (K.west);
  \draw[->] (K.east) -- (G.west) node[above left]{$u$};
  \draw[->] (G.east) -- (adddy.west);
  \draw[->] ($(G.west)+(-0.8, 0)$) node[branch](cffb){} |- (Gm.east);
  \draw[->] (Gm.west) -- (Hh.east);
  \draw[->] (Hh.west) -| (addK.south);
  \draw[<-] (addn.east) -- ++(\cdist, 0) coordinate[](endpos) node[above left]{$n$};
  \draw[->] (adddy.east) -- (G-|endpos) node[above left]{$y$};
  \draw[->] (adddy-|addn) node[branch]{} -- (addn.north);
  \draw[<-] (addfb.west) -- ++(-\cdist, 0) node[above right](r){$r$};
  \draw[->] (addn.west) -- (Hl.east) node[above right]{$y_m$};
  \draw[<-] (adddy.north) -- ++(0, \cdist) node[below right]{$d_y$};
  \draw[->] (Hl.west) -| (addfb.south) node[below right]{};

  \begin{scope}[on background layer]
    \node[fit={(Hl.south -| cffb) (r.north west)}, inner sep=8pt, draw, fill=black!20!white, dashed] (Kfb) {};
  \end{scope}

  \begin{scope}[on background layer]
    \node[fit={($(addK.west|-Hh.south)+(-0.1, 0)$) (K.north-|cffb)}, inner sep=4pt, fill=colorblue!20!white, dashed] (Kfb) {};
  \end{scope}
\end{tikzpicture}
#+end_src

#+name: fig:detail_control_cf_arch_eq
#+caption: Equivalent feedback architecture
#+RESULTS:
[[file:figs/detail_control_cf_arch_eq.png]]

The dynamics of the system can be rewritten eqref:eq:detail_control_cf_sf_cl_tf_K with $K = \frac{k}{1 + H_H G k}$.

\begin{subequations}\label{eq:detail_control_cf_sf_cl_tf_K}
  \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}
\end{subequations}

**** Asymptotic behavior

Let's take the extreme case of very high values for $k$.
In that case $K(s)$ converges to plant inverse multiply by the inverse of the high pass filter eqref:eq:detail_control_cf_high_k.

\begin{equation}\label{eq:detail_control_cf_high_k}
  \lim_{k\to\infty} K(s) = \lim_{k\to\infty} \frac{k}{1+H_H(s) G(s) k} = \left( H_H(s) G(s) \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.

With this assumptions, the resulting control architecture is shown on Figure ref:fig:detail_control_cf_arch_class.
The only "tuning parameters" are the complementary filters $H_L$ and $H_H$.

#+begin_src latex :file detail_control_cf_arch_class.pdf
\tikzset{block/.default={0.8cm}{0.6cm}}
\tikzset{addb/.append style={scale=0.7}}
\tikzset{node distance=0.6}
\def\cdist{0.7}

\begin{tikzpicture}
  \node[addb={+}{}{}{}{-}] (addfb) at (0, 0){};
  \node[block, right=of addfb] (K){$K$};
  \node[block, right=of K] (G){$G^\prime$};
  \node[addb={+}{}{}{}{}, right=of G] (adddy){};
  \node[addb={+}{}{}{}{}, below right=0.7 and 0.3 of adddy] (addn) {};
  \node[block] (Hl) at (K|-addn) {$H_L$};

  \draw[->] (addfb.east) -- (K.west) node[above left]{};
  \draw[->] (K.east) -- (G.west) node[above left]{$u$};
  \draw[->] (G.east) -- (adddy.west);
  \draw[<-] (addn.east) -- ++(\cdist, 0) coordinate[](endpos) node[above left]{$n$};
  \draw[->] (G-|addn)node[branch]{} -- (addn.north);
  \draw[->] (adddy.east) -- (G-|endpos) node[above left]{$y$};
  \draw[<-] (addfb.west) -- ++(-\cdist, 0) node[above right](r){$r$};
  \draw[->] (addn.west) -- (Hl.east);
  \draw[->] (Hl.west) -| (addfb.south);
  \draw[<-] (adddy.north) -- ++(0, \cdist) node[below right]{$d_y$};

  \begin{scope}[on background layer]
    \node[fit={(Hl.south east) (r.north west)}, inner sep=4pt, draw, fill=black!20!white, dashed, label={RT controller}] (Kfb) {};
  \end{scope}
\end{tikzpicture}
#+end_src

#+name: fig:detail_control_cf_arch_class
#+caption: Equivalent classical feedback control architecture
#+RESULTS:
[[file:figs/detail_control_cf_arch_class.png]]

The equations describing the dynamics of the closed-loop system of Figure ref:fig:detail_control_cf_arch_class

\begin{subequations}
  \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_cf_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_cf_cl_system_u}
  \end{align}
\end{subequations}

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 the closed loop transfer functions are described by eqref:eq:detail_control_cf_sf_cl_tf_K_inf.

\begin{subequations}\label{eq:detail_control_cf_sf_cl_tf_K_inf}
  \begin{alignat}{5}
    y &=  H_H        dy &&+         r &&- H_L         n \label{eq:detail_control_cf_cl_performance_y} \\
    u &= -G^{-1} H_L dy &&+ G^{-1}  r &&- G^{-1} H_L  n \label{eq:detail_control_cf_cl_performance_u}
  \end{alignat}
\end{subequations}

The obtained sensitivity transfer function equals to the high pass filter $S = \frac{y}{dy} = H_H$ and the 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.

** Translating the performance requirements into the shapes of the complementary filters
 <<ssec:detail_control_cf_trans_perf>>
**** Introduction                                                  :ignore:
The required performance specifications in a feedback system can usually be translated into requirements on the upper bounds of $|S(j\omega)|$ and $|T(j\omega)|$ [[cite:&bibel92_guidel_h]].
The process of designing a controller $K(s)$ in order to obtain the desired shapes of $|S(j\omega)|$ and $|T(j\omega)|$ is called closed-loop shaping.

The equations eqref:eq:detail_control_cf_cl_system_y and eqref:eq:detail_control_cf_cl_system_u describing the dynamics of the studied feedback architecture are not written in terms of the controller $K(s)$ but in terms of the complementary filters $H_L(s)$ and $H_H(s)$.
The typical specifications are then translated into the desired shapes of the complementary filters.

**** Nominal Stability (NS)
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 ($G^\prime = G$), the sensitivity transfer function is equal to the high pass filter: $S(s) = H_H(s)$.

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).
Therefore stable and minimum phase complementary filters need to be used.

**** Nominal Performance (NP)
Two performance weights $w_H$ and $w_L$ are here defined in such a way that performance specifications are satisfied is eqref:eq:detail_control_cf_weights is satisfied.

\begin{subequations}\label{eq:detail_control_cf_weights}
  \begin{align}
    |w_H(j\omega) S(j\omega)| &\le 1 \quad \forall\omega\\
    |w_L(j\omega) T(j\omega)| &\le 1 \quad \forall\omega
  \end{align}
\end{subequations}

For the nominal system, we have $S = H_H$ and $T = H_L$, and then nominal performance is ensured by requiring eqref:eq:detail_control_cf_nominal_performance.

\begin{empheq}[box=\fbox,left=\text{NP} \Longleftrightarrow \empheqlbrace]{align} \label{eq:detail_control_cf_nominal_performance}
  &|w_H(j\omega) H_H(j\omega)| \le 1 \quad \forall\omega \label{eq:detail_control_cf_nominal_perf_hh}\\
  &|w_L(j\omega) H_L(j\omega)| \le 1 \quad \forall\omega \label{eq:detail_control_cf_nominal_perf_hl}
\end{empheq}

The translation of typical performance requirements on the shapes of the complementary filters is discussed below:
- for disturbance rejections, make $|S| = |H_H|$ small
- for noise attenuation, make $|T| = |H_L|$ small
- closed-loop bandwidth can be limited by requiring that $|T|$ is less than $\frac{1}{\sqrt{2}}$ above the maximum wanted bandwidth

Therefore, by properly choosing the shape of the complementary filters, the nominal performance specifications can be addressed.

Classical stability margins (gain and phase margins) can also be linked to the maximum amplitude of the sensitivity transfer function.
- [ ] Add reference

Typically, having $|S|_{\infty} \le 2$ guarantees a gain margin of at least $2$ and a phase margin of at least $\SI{29}{\degree}$.

**** TODO Response time to change of reference signal
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_cf_cl_performance_y and does not depends of the complementary filters.

However, one can add a pre-filter as shown in Figure ref:fig:detail_control_cf_arch_class_prefilter.

#+begin_src latex :file detail_control_cf_arch_class_prefilter.pdf
\tikzset{block/.default={0.8cm}{0.6cm}}
\tikzset{addb/.append style={scale=0.7}}
\tikzset{node distance=0.6}
\def\cdist{0.7}

\begin{tikzpicture}
  \node[addb={+}{}{}{}{-}] (addfb) at (0, 0){};
  \node[block, left=of addfb] (Kr){$K_r$};
  \node[block, right=of addfb] (K){$K$};
  \node[block, right=of K] (G){$G^\prime$};
  \node[addb={+}{}{}{}{}, right=of G] (adddy){};
  \node[addb={+}{}{}{}{}, below right=0.7 and 0.3 of adddy] (addn) {};
  \node[block, left=of addn] (Hl) {$H_L$};

  \draw[->] (addfb.east) -- (K.west) node[above left]{};
  \draw[->] (K.east) -- (G.west) node[above left]{$u$};
  \draw[->] (G.east) -- (adddy.west);
  \draw[<-] (addn.east) -- ++(\cdist, 0) coordinate[](endpos) node[above left]{$n$};
  \draw[->] (G-|addn)node[branch]{} -- (addn.north);
  \draw[->] (adddy.east) -- (G-|endpos) node[above left]{$y$};
  \draw[<-] (Kr.west) -- ++(-\cdist, 0) node[above right]{$r$};
  \draw[->] (Kr.east) -- (addfb.west);
  \draw[->] (addn.west) -- (Hl.east);
  \draw[->] (Hl.west) -| (addfb.south);
  \draw[<-] (adddy.north) -- ++(0, \cdist) node[below right]{$d_y$};
\end{tikzpicture}
#+end_src

#+name: fig:detail_control_cf_arch_class_prefilter
#+caption: Prefilter used to limit input usage
#+RESULTS:
[[file:figs/detail_control_cf_arch_class_prefilter.png]]

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.

**** TODO Input usage
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 (Figure ref:fig:detail_control_cf_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.

**** Robust Stability (RS)

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 for the design of 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, the  input multiplicative uncertainty as represented in Figure ref:fig:detail_control_cf_input_uncertainty is used.

Then, the set of possible plant is described by eqref:eq:detail_control_cf_multiplicative_uncertainty.
$w_I$ should be chosen such that all possible plants $G^\prime$ are contained in the set $\Pi_i$.

\begin{equation}\label{eq:detail_control_cf_multiplicative_uncertainty}
  \Pi_i: \quad G^\prime(s) = G(s)\big(1 + w_I(s)\Delta_I(s)\big); \quad |\Delta_I(j\omega)| \le 1 \ \forall\omega
\end{equation}

#+begin_src latex :file detail_control_cf_input_uncertainty.pdf
\tikzset{block/.default={0.8cm}{0.6cm}}
\tikzset{addb/.append style={scale=0.7}}
\tikzset{node distance=0.6}
\def\cdist{0.7}

\begin{tikzpicture}
  % Blocs
  \node[block] (G) {$G$};

  \node[addb, left= of G] (addi) {};
  \node[block, above left=0.3 and 0.3 of addi] (deltai) {$\Delta_I$};
  \node[block, left= of deltai] (wi) {$w_I$};
  \node[branch] (branch) at ($(wi.west|-addi)+(-0.4, 0)$) {};

  % Connections and labels
  \draw[->] (branch.center) |- (wi.west);
  \draw[->] ($(branch)+(-0.6, 0)$) -- (addi.west);
  \draw[->] (wi.east) -- (deltai.west);
  \draw[->] (deltai.east) -| (addi.north);
  \draw[->] (addi.east) -- (G.west);
  \draw[->] (G.east) -- ++(0.6, 0);

  \begin{scope}[on background layer]
    \node[fit={(branch|-wi.north) (G.south east)}, inner sep=6pt, draw, dashed, fill=black!20!white] (Gp) {};
    \node[below left] at (Gp.north east) {$G\prime$};
  \end{scope}
\end{tikzpicture}
#+end_src

#+name: fig:detail_control_cf_input_uncertainty_nyquist
#+caption: Input multiplicative uncertainty to model the differences between the model and the physical plant (\subref{fig:detail_control_cf_input_uncertainty}). Effect of this uncertainty is displayed on the Nyquist plot (\subref{fig:detail_control_cf_nyquist_uncertainty})
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:detail_control_cf_input_uncertainty}Input multiplicative uncertainty}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
#+attr_latex: :scale 1
[[file:figs/detail_control_cf_input_uncertainty.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_control_cf_nyquist_uncertainty}Nyquist plot - Effect of multiplicative uncertainty}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
#+attr_latex: :scale 1
[[file:figs/detail_control_cf_nyquist_uncertainty.png]]
#+end_subfigure
#+end_figure

Considering input multiplicative uncertainty, the robust stability property can be derived graphically from the Nyquist plot (Figure ref:fig:detail_control_cf_nyquist_uncertainty), and eqref:eq:detail_control_cf_robust_stability_graphically is obtained, as proposed in [[cite:&skogestad07_multiv_feedb_contr, chapt. 7.5.1]].


\begin{equation}\label{eq:detail_control_cf_robust_stability_graphically}
    \text{RS} \Longleftrightarrow \left|w_I(j\omega) L(j\omega) \right| \le \left| 1 + L(j\omega) \right| \quad \forall\omega
\end{equation}

After some algebraic manipulations, robust stability is then guaranteed by having the low pass filter $H_L$ satisfying eqref:eq:detail_control_cf_condition_robust_stability.

\begin{equation}\label{eq:detail_control_cf_condition_robust_stability}
  \boxed{\text{RS} \Longleftrightarrow |w_I(j\omega) H_L(j\omega)| \le 1 \quad \forall \omega}
\end{equation}

**** Robust Performance (RP)
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 eqref:eq:detail_control_cf_robust_perf_S.

\begin{equation}\label{eq:detail_control_cf_robust_perf_S}
  \text{RP} \Longleftrightarrow |w_H(j\omega) S(j\omega)| \le 1 \quad \forall G^\prime \in \Pi_I, \ \forall\omega
\end{equation}

Let's transform condition eqref:eq:detail_control_cf_robust_perf_S into a condition on the complementary filters eqref:eq:detail_control_cf_robust_performance.

\begin{equation}\label{eq:detail_control_cf_robust_performance}
  \boxed{\text{RP} \Longleftrightarrow | w_H(j\omega) H_H(j\omega) | + | w_I(j\omega) H_L(j\omega) | \le 1, \ \forall\omega}
\end{equation}

Obtained condition for robust performance combines both the NP and RS conditions.
If both NP and RS conditions are fulfilled, the robust performance will be fulfilled with a factor 2 [[cite:&skogestad07_multiv_feedb_contr, chapt. 7.6]].
Therefore, for SISO systems, robust stability and nominal performance are usually sufficient.

** Complementary filter design
<<ssec:detail_control_cf_analytical_complementary_filters>>

As was explained in Section ref:sec:detail_control_sensor, complementary filters can easily be shaped with the standard $\mathcal{H}_{\infty}$ synthesis.
As requirements can usually be expressed as upper bounds on the complementary filters' magnitude, this method is very well suited.

However, analytical formulas for complementary filters may also be used.

For some applications, first order complementary filters eqref:eq:detail_control_cf_1st_order are sufficient.

\begin{subequations}\label{eq:detail_control_cf_1st_order}
  \begin{align}
    H_L(s) &= \frac{1}{1 + s/\omega_0} \\
    H_H(s) &= \frac{s/\omega_0}{1 + s/\omega_0}
  \end{align}
\end{subequations}

They can be expressed analytically in the digital domain using the Bilinear transformation.
In such case, digital filters eqref:eq:detail_control_cf_1st_order_z are obtained.

\begin{subequations}\label{eq:detail_control_cf_1st_order_z}
  \begin{align}
    H_L(z^{-1}) &= \frac{T_s \omega_0 + T_s \omega_0 z^{-1}}{T_s \omega_0 + 2 + (T_s \omega_0 - 2) z^{-1}} \\
    H_H(z^{-1}) &= \frac{2 - 2 z^{-1}}{T_s \omega_0 + 2 + (T_s \omega_0 - 2) z^{-1}}
  \end{align}
\end{subequations}

The main advantage of having analytical formulas for the complementary filters is that the parameter $\omega_0$ may be modified in real time.
This is illustrated in Figure ref:fig:detail_control_cf_arch_tunable_params.
Therefore, the performance and robustness of different control bandwidth can be tested very quickly.

#+begin_src latex :file detail_control_cf_arch_tunable_params.pdf
\tikzset{block/.default={0.8cm}{0.6cm}}
\tikzset{addb/.append style={scale=0.7}}
\tikzset{node distance=0.6}
\def\cdist{0.7}

\begin{tikzpicture}
  \node[addb={+}{}{}{}{-}] (addfb) at (0, 0){};
  \node[block, right=of addfb] (Hh){$H_H^{-1}$};
  \node[block, right=of Hh] (Ginv){$G^{-1}$};
  \node[block, right=of Ginv] (G){$G^\prime$};
  \node[addb={+}{}{}{}{}, right=of G] (adddy){};
  \node[addb={+}{}{}{}{}, below right=1.2 and 0.3 of adddy] (addn) {};
  \node[block] (Hl) at (Hh|-addn) {$H_L$};

  \node[color=colorred] (wb) at ($0.5*(Hh.south) + 0.5*(Hl.north)$) {$\bullet$};
  \draw[-, color=colorred] ($(wb) + (-0.6, 0)$)node[left]{$\omega_0$} -- (wb.center);
  \draw[->, color=colorred] (wb.center) -- (Hh.south);
  \draw[->, color=colorred] (wb.center) -- (Hl.north);

  \draw[->] (addfb.east) -- (Hh.west);
  \draw[->] (Hh.east) -- (Ginv.west);
  \draw[->] (Ginv.east) -- (G.west) node[above left]{$u$};
  \draw[->] (G.east) -- (adddy.west);
  \draw[<-] (addn.east) -- ++(\cdist, 0) coordinate[](endpos) node[above left]{$n$};
  \draw[->] (G-|addn)node[branch]{} -- (addn.north);
  \draw[->] (adddy.east) -- (G-|endpos) node[above left]{$y$};
  \draw[<-] (addfb.west) -- ++(-\cdist, 0) node[above right](r){$r$};
  \draw[->] (addn.west) -- (Hl.east);
  \draw[->] (Hl.west) -| (addfb.south);
  \draw[<-] (adddy.north) -- ++(0, \cdist) node[below right]{$d_y$};

  \begin{scope}[on background layer]
    \node[fit={(Hl.south -| Ginv.east) (r.north west)}, inner sep=4pt, draw, fill=black!20!white, dashed, label={RT controller}] (Kfb) {};
  \end{scope}
\end{tikzpicture}
#+end_src

#+name: fig:detail_control_cf_arch_tunable_params
#+caption: Implemented digital complementary filters with parameter $\omega_0$ that can be changed in real time
#+RESULTS:
[[file:figs/detail_control_cf_arch_tunable_params.png]]

For many applications, slope of $+2$ is wanted at low frequency for the sensitivity transfer function (to follow ramp inputs for instance), and a slope of $-2$ for the complementary sensitivity transfer function.
In that case, complementary filters shown in equation eqref:eq:detail_control_cf_2nd_order are proposed.

\begin{subequations}\label{eq:detail_control_cf_2nd_order}
  \begin{align}
    H_L(s) &= \frac{(1+\alpha) (\frac{s}{\omega_0})+1}{\left((\frac{s}{\omega_0})+1\right) \left((\frac{s}{\omega_0})^2 + \alpha (\frac{s}{\omega_0}) + 1\right)}\\
    H_H(s) &= \frac{(\frac{s}{\omega_0})^2 \left((\frac{s}{\omega_0})+1+\alpha\right)}{\left((\frac{s}{\omega_0})+1\right) \left((\frac{s}{\omega_0})^2 + \alpha (\frac{s}{\omega_0}) + 1\right)}
  \end{align}
\end{subequations}

The effect of $\alpha$ and $\omega_0$ and the obtained shape of the complementary filters is shown in Figure ref:fig:detail_control_cf_analytical_effect.
Such filters can also be implemented in the digital domain with analytical formulas, such as $\alpha$ and $\omega_0$ can be changed in real time.

#+begin_src matlab :exports none :results none
%% Analytical Complementary Filters - Effect of alpha
freqs_study = logspace(-2, 2, 1000);
alphas = [0.1, 1, 10];
w0 = 2*pi*1;
s = tf('s')

figure;
hold on;
for i = 1:length(alphas)
    alpha = alphas(i);
    Hh2 = (s/w0)^2*((s/w0)+1+alpha)/(((s/w0)+1)*((s/w0)^2 + alpha*(s/w0) + 1));
    Hl2 = ((1+alpha)*(s/w0)+1)/(((s/w0)+1)*((s/w0)^2 + alpha*(s/w0) + 1));
    plot(freqs_study, abs(squeeze(freqresp(Hh2, freqs_study, 'Hz'))), 'color', colors(i,:), 'DisplayName', sprintf('$\\alpha = %g$', alphas(i)));
    plot(freqs_study, abs(squeeze(freqresp(Hl2, freqs_study, 'Hz'))), 'color', colors(i,:), 'HandleVisibility', 'off');
end
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Relative Frequency $\frac{\omega}{\omega_0}$'); ylabel('Magnitude');
hold off;
ylim([1e-3, 20]);
leg = legend('location', 'northeast', 'FontSize', 8);
leg.ItemTokenSize(1) = 18;
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/detail_control_cf_analytical_effect_alpha.pdf', 'width', 'half', 'height', 'normal');
#+end_src

#+begin_src matlab :exports none :results none
%% Analytical Complementary Filters - Effect of w0
freqs_study = logspace(-1, 3, 1000);
alpha = [1];
w0s = [2*pi*1, 2*pi*10, 2*pi*100];
s = tf('s')

figure;
hold on;
for i = 1:length(w0s)
    w0 =w0s(i);
    Hh2 = (s/w0)^2*((s/w0)+1+alpha)/(((s/w0)+1)*((s/w0)^2 + alpha*(s/w0) + 1));
    Hl2 = ((1+alpha)*(s/w0)+1)/(((s/w0)+1)*((s/w0)^2 + alpha*(s/w0) + 1));
    plot(freqs_study, abs(squeeze(freqresp(Hh2, freqs_study, 'Hz'))), 'color', colors(i,:), 'DisplayName', sprintf('$\\omega_0 = %g$ Hz', w0/2/pi));
    plot(freqs_study, abs(squeeze(freqresp(Hl2, freqs_study, 'Hz'))), 'color', colors(i,:), 'HandleVisibility', 'off');
end
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
hold off;
xlim([freqs_study(1), freqs_study(end)]); ylim([1e-3, 20]);
leg = legend('location', 'southeast', 'FontSize', 8);
leg.ItemTokenSize(1) = 18;
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/detail_control_cf_analytical_effect_w0.pdf', 'width', 'half', 'height', 'normal');
#+end_src

#+name: fig:detail_control_cf_analytical_effect
#+caption: Shape of proposed analytical complementary filters. Effect of $\alpha$ (\subref{fig:detail_control_cf_analytical_effect_alpha}) and $\omega_0$ (\subref{fig:detail_control_cf_analytical_effect_w0}) are shown.
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:detail_control_cf_analytical_effect_alpha}Effect of $\alpha$}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
#+attr_latex: :width 0.95\linewidth
[[file:figs/detail_control_cf_analytical_effect_alpha.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_control_cf_analytical_effect_w0}Effect of $\omega_0$}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
#+attr_latex: :width 0.95\linewidth
[[file:figs/detail_control_cf_analytical_effect_w0.png]]
#+end_subfigure
#+end_figure

** Numerical Example
<<ssec:detail_control_cf_simulations>>
**** Procedure

In order to apply this control technique, we propose the following procedure:
1. Identify the plant to be controlled in order to obtain the plant model $G$
2. Design the weighting function $w_I$ such that all possible plants $G^\prime$ are contained in the set $\Pi_i$
3. Translate the performance requirements into upper bounds on the complementary filters (as explained in Section ref:ssec:detail_control_cf_trans_perf)
4. Design the weighting functions $w_H$ and $w_L$ and generate the complementary filters using $\mathcal{H}_{\infty}\text{-synthesis}$ (as was explained in Section ref:ssec:detail_control_sensor_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.
   For simple cases, analytical formulas of complementary filters given in Section ref:ssec:detail_control_cf_analytical_complementary_filters can be used.
6. If $K(s) = \left( G(s) H_H(s) \right)^{-1}$ is not proper, low pass filters should be added high a high corner frequency

**** Plant

- To test this control architecture, a simple test model is used (Figure ref:fig:detail_control_cf_test_model).
- This model is quite similar to many positioning stages for Synchrotrons.
  A payload (i.e. sample) with mass $m = 5$ is positioned on top of the stage.
  The goal is to position the sample with respect to the x-ray.
  It is supposed that the relative position $y$ between the payload an the x-ray is measured (typically the relative position between the focusing optics and the sample is performed).
  There are some disturbance forces acting on the positioning stability, such as stage vibration $d_w$ and direct forces applied on the sample $d_F$ (for instance cable forces).
  The positioning stage itself has a stiffness $k$, an internal damping $c$ and the force $F$ can be controlled.

The model of the plant $G(s)$ from actuator force $F$ to displacement $y$ is then

\begin{equation}\label{eq:detail_control_cf_test_plant_tf}
  G(s) = \frac{1}{m s^2 + c s + k}
\end{equation}

The values for the parameters of the models are $m = \SI{20}{\kg}$, $k = 1\si{\N/\mu\m}$ and $c = 10^2\si{\N\per(\m\per\s)}$.

The plant dynamics has some uncertainty related to the limited support compliance, unmodeled flexible dynamics, dynamics of the payload, etc.
A multiplicative input uncertainty weight $w_I(s)$ is used to specify the amount of uncertainty as a function of frequency eqref:eq:detail_control_cf_test_plant_uncertainty.

\begin{equation}\label{eq:detail_control_cf_test_plant_uncertainty}
  w_I(s) = 10 \cdot \frac{(s+100)^2}{(s+1000)^2}
\end{equation}

The nominal plant dynamics as well as the entire set of possible plants $\Pi_i$ are shown in Figure ref:fig:detail_control_cf_bode_plot_mech_sys.
#+begin_src matlab
%% Test model
freqs = logspace(0, 3, 1000); % Frequency Vector [Hz]

m = 20;  % mass [kg]
k = 1e6; % stiffness [N/m]
c = 1e2; % damping [N/(m/s)]

% Plant dynamics
G = 1/(m*s^2 + c*s + k);

% Uncertainty weight
wI = generateWF('n', 2, 'w0', 2*pi*50, 'G0', 0.1, 'Ginf', 10, 'Gc', 1);
#+end_src

#+begin_src matlab :exports none :results none
%% Bode plot of the plant with dynamical uncertainty
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');

% Magnitude
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(G, freqs, 'Hz'))), 'k-', 'DisplayName', 'G');
plotMagUncertainty(wI, freqs, 'G', G, 'DisplayName', '$\Pi_i$');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Magnitude [m/N]'); set(gca, 'XTickLabel',[]);
ylim([1e-8, 7e-5]);
hold off;
leg = legend('location', 'northeast', 'FontSize', 8);
leg.ItemTokenSize(1) = 18;

% Phase
ax2 = nexttile;
hold on;
plotPhaseUncertainty(wI, freqs, 'G', G);
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G, freqs, 'Hz')))), 'k-');
set(gca,'xscale','log');
yticks(-360:90:90);
ylim([-270 45]);
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;

linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/detail_control_cf_bode_plot_mech_sys.pdf', 'width', 'half', 'height', 450);
#+end_src

#+name: fig:detail_control_cf_test_model_plant
#+caption: Schematic of the test system (\subref{fig:detail_control_cf_test_model}). Bode plot of the transfer function $G(s)$ from $F$ to $y$ and the associated uncertainty set (\subref{fig:detail_control_cf_bode_plot_mech_sys}).
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:detail_control_cf_test_model}Test model}
#+attr_latex: :options {0.3\textwidth}
#+begin_subfigure
#+attr_latex: :scale 1
[[file:figs/detail_control_cf_test_model.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_control_cf_bode_plot_mech_sys}Bode plot of $G(s)$ and associated uncertainty set}
#+attr_latex: :options {0.66\textwidth}
#+begin_subfigure
#+attr_latex: :scale 1
[[file:figs/detail_control_cf_bode_plot_mech_sys.png]]
#+end_subfigure
#+end_figure

**** Requirements and choice of complementary filters

As explained in Section ref:ssec:detail_control_cf_trans_perf, nominal performance requirements can be expressed as upper bounds on the complementary filter shapes.
- Be able to follow ramp inputs (i.e. constant velocity scans) with no steady-state tracking error.
  This requires to have a $+2$ slope at low frequency for $|S(j\omega)|$
- Filter the measurement noise above 300Hz as the sensor noise is high (say a filtering factor of 100 is needed above that frequency).
- As much disturbance rejection as possible.

The second requirement is to have robust stability meaning that the plant should remain stable while considering the dynamical uncertainties modelled with $w_I$
- The low-pass complementary filter magnitude $|H_L(j\omega)|$ should be bellow the inverse magnitude of the uncertainty weight magnitude $|w_I(j\omega)|$ eqref:eq:detail_control_cf_condition_robust_stability

Robust performance is ensured by simultaneous NP and RS.

All the requirements on $H_L$ and $H_H$ are represented on Figure ref:fig:detail_control_cf_specs_S_T.

#+name: fig:detail_control_cf_specs_S_T_obtained_filters
#+caption: Performance requirement and complementary filters used (\subref{fig:detail_control_cf_specs_S_T}). Obtained controller from the complementary filters and the plant inverse is shown in (\subref{fig:detail_control_cf_bode_Kfb}).
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:detail_control_cf_specs_S_T}Specifications and complementary filters}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
#+attr_latex: :width 0.95\linewidth
[[file:figs/detail_control_cf_specs_S_T.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_control_cf_bode_Kfb}Bode plot of $K(s) \cdot H_L(s)$}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
#+attr_latex: :width 0.95\linewidth
[[file:figs/detail_control_cf_bode_Kfb.png]]
#+end_subfigure
#+end_figure

While the $\mathcal{H}_\infty$ synthesis of complementary filters could be used, for this simple examples with simple requirements, analytical formulas of complementary filters were used.

For this simple example, analytical formulas proposed to have +2 and -2 slopes eqref:eq:detail_control_cf_2nd_order were used.
$\alpha = 1$ and $\omega_0 = 2\pi \cdot 20$ were used.

#+begin_src matlab
%% Analytical Complementary Filters
w0 = 2*pi*20;
alpha = 1;

Hh = (s/w0)^2*((s/w0)+1+alpha)/(((s/w0)+1)*((s/w0)^2 + alpha*(s/w0) + 1));
Hl = ((1+alpha)*(s/w0)+1)/(((s/w0)+1)*((s/w0)^2 + alpha*(s/w0) + 1));
#+end_src

#+begin_src matlab :exports none :results none
%% Specifications
figure;
hold on;
plot([1, 100], [0.01, 100], ':', 'color', colors(2,:));
plot([300, 1000], [0.01, 0.01], ':', 'color', colors(1,:));
plot(freqs, 1./abs(squeeze(freqresp(wI, freqs, 'Hz'))), ':', 'color', colors(1,:));
plot(freqs, abs(squeeze(freqresp(Hl, freqs, 'Hz'))), '-', 'color', colors(1,:));
plot(freqs, abs(squeeze(freqresp(Hh, freqs, 'Hz'))), '-', 'color', colors(2,:));
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
hold off;
xlim([freqs(1), freqs(end)]);
ylim([1e-3, 10]);
xticks([0.1, 1, 10, 100, 1000]);
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
% exportFig('figs/detail_control_cf_specs_S_T.pdf', 'width', 'half', 'height', 'normal');
#+end_src

**** Controller analysis

The controller to be implemented is $K(s) = \tilde{G}^{-1}(s) H_H^{-1}(s)$, with $\tilde{G}^{-1}(s)$ is the plant inverse which needs to be stable and proper.
Therefore, some low pass filters are added at high frequency eqref:eq:detail_control_cf_test_plant_inverse.

\begin{equation}\label{eq:detail_control_cf_test_plant_inverse}
  \tilde{G}^{-1}(s) = \frac{m s^2 + c s + k}{1 + \frac{s}{2\pi \cdot 1000} + \left( \frac{s}{2\pi \cdot 1000} \right)^2}
\end{equation}

The obtained bode plot of the controller times the complementary high pass filter is shown in Figure ref:fig:detail_control_cf_bode_Kfb.
- two integrators are present at low frequency to be able to follow ramp inputs
- a notch is located at the plant resonance (inverse)
- a lead is added near the bandwidth around $\SI{20}{Hz}$

#+begin_src matlab
%% Obtained controller
omega = 2*pi*1000;

K = 1/(Hh*G) * 1/((1+s/omega+(s/omega)^2));
K = zpk(minreal(K));
#+end_src

#+begin_src matlab :exports none :results none
%% Bode plot of the controller K
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');

% Magnitude
ax1 = nexttile([2, 1]);
plot(freqs, abs(squeeze(freqresp(K*Hl, freqs, 'Hz'))), 'k-');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
ylim([8e3, 1e8])

% Phase
ax2 = nexttile;
plot(freqs, 180/pi*angle(squeeze(freqresp(K*Hl, freqs, 'Hz'))), 'k-');
set(gca,'xscale','log');
yticks(-180:45:180);
ylim([-180 45]);
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');

linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/detail_control_cf_bode_Kfb.pdf', 'width', 'half', 'height', 500);
#+end_src

**** Robustness and Performance analysis
The robust stability can be access on the Nyquist plot (Figure ref:fig:detail_control_cf_nyquist_robustness).
Even when considering all the possible plants in the uncertainty set, the nyquist plot stays away from the unstable point, indicating good robustness.

The performance is evaluated by looking at the closed-loop sensitivity and complementary sensitivity transfer functions (Figure ref:fig:detail_control_cf_robust_perf).

#+begin_src matlab
num_delta_points = 50;
theta = linspace(0, 2*pi, num_delta_points);
delta_points = exp(1j * theta);

% Get frequency responses for all components
G_resp  = squeeze(freqresp(G,  freqs, 'Hz'));
K_resp  = squeeze(freqresp(K,  freqs, 'Hz'));
Hl_resp = squeeze(freqresp(Hl, freqs, 'Hz'));
wI_resp = squeeze(freqresp(wI, freqs, 'Hz'));

% Calculate nominal responses
nom_L = G_resp .* K_resp .* Hl_resp;
nom_S = 1 ./ (1 + nom_L);
nom_T = nom_L ./ (1 + nom_L);

% Store all the points in the complex plane that L can take
loop_region_points = zeros(length(freqs), num_delta_points);

% Initialize arrays to store magnitude bounds
S_mag_min = ones(length(freqs), 1) * inf;
S_mag_max = zeros(length(freqs), 1);
T_mag_min = ones(length(freqs), 1) * inf;
T_mag_max = zeros(length(freqs), 1);

% Calculate magnitude bounds for all delta values
for i = 1:num_delta_points
    % Perturbed loop gain
    loop_perturbed = nom_L .* (1 + wI_resp .* delta_points(i));
    loop_region_points(:,i) = loop_perturbed;

    % Perturbed sensitivity function
    S_perturbed = 1 ./ (1 + loop_perturbed);
    S_mag = abs(S_perturbed);

    % Update S magnitude bounds
    S_mag_min = min(S_mag_min, S_mag);
    S_mag_max = max(S_mag_max, S_mag);

    % Perturbed complementary sensitivity function
    T_perturbed = loop_perturbed ./ (1 + loop_perturbed);
    T_mag = abs(T_perturbed);

    % Update T magnitude bounds
    T_mag_min = min(T_mag_min, T_mag);
    T_mag_max = max(T_mag_max, T_mag);
end

% At frequencies where |wI| > 1, T min is zero
T_mag_min(abs(wI_resp)>1) = 1e-10;
#+end_src

#+begin_src matlab
%% Nyquist plot to check Robust Stability
figure;
hold on;
plot(real(squeeze(freqresp(G*K*Hl, freqs, 'Hz'))), imag(squeeze(freqresp(G*K*Hl, freqs, 'Hz'))), 'k', 'DisplayName', '$L(j\omega)$ - Nominal');
plot(alphaShape(real(loop_region_points(:)), imag(loop_region_points(:)), 0.1), 'FaceColor', [0, 0, 0], 'EdgeColor', 'none', 'FaceAlpha', 0.3, 'DisplayName', '$L(j\omega)$ - $\forall G \in \Pi_i$');
plot(-1, 0, 'k+', 'MarkerSize', 5, 'HandleVisibility', 'off');
hold off;
grid on;
axis equal
xlim([-1.4, 0.2]); ylim([-1.2, 0.4]);
xticks(-1.4:0.2:0.2); yticks(-1.2:0.2:0.4);
xlabel('Real Part'); ylabel('Imaginary Part');
leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 18;
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/detail_control_cf_nyquist_robustness', 'width', 'half', 'height', 'normal');
#+end_src

#+begin_src matlab :exports none :results none
%% Robust Performance
figure;
hold on;
plot(freqs, abs(nom_S), 'color', colors(2,:), 'DisplayName', '$|S|$ - Nom.');
plot(freqs, abs(nom_T), 'color', colors(1,:), 'DisplayName', '$|T|$ - Nom.');

patch([freqs, fliplr(freqs)], [S_mag_max', fliplr(S_mag_min')], colors(2,:), 'FaceAlpha', 0.2, 'EdgeColor', 'none', 'HandleVisibility', 'off');
patch([freqs, fliplr(freqs)], [T_mag_max', fliplr(T_mag_min')], colors(1,:), 'FaceAlpha', 0.2, 'EdgeColor', 'none', 'HandleVisibility', 'off');

plot([1, 100], [0.01, 100], ':', 'color', colors(2,:), 'DisplayName', 'Specs.');
plot([300, 1000], [0.01, 0.01], ':', 'color', colors(1,:), 'DisplayName', 'Specs.');
plot(freqs, 1./abs(squeeze(freqresp(wI, freqs, 'Hz'))), ':', 'color', colors(1,:), 'HandleVisibility', 'off');

set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
hold off;
xlabel('Frequency [Hz]'); ylabel('Magnitude');
xlim([freqs(1), freqs(end)]);
ylim([1e-4, 5]);
xticks([0.1, 1, 10, 100, 1000]);
leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 3);
leg.ItemTokenSize(1) = 18;
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/detail_control_cf_robust_perf.pdf', 'width', 'half', 'height', 'normal');
#+end_src

#+name: fig:detail_control_cf_simulation_results
#+caption: Validation of Robust stability with the Nyquist plot (\subref{fig:detail_control_cf_nyquist_robustness}) and validation of the nominal and robust performance with the magnitude of the closed-loop transfer functions (\subref{fig:detail_control_cf_robust_perf})
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:detail_control_cf_nyquist_robustness}Robust Stability}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
#+attr_latex: :scale 0.8
[[file:figs/detail_control_cf_nyquist_robustness.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_control_cf_robust_perf}Nominal and Robust performance}
#+attr_latex: :options {0.49\textwidth}
#+begin_subfigure
#+attr_latex: :scale 0.8
[[file:figs/detail_control_cf_robust_perf.png]]
#+end_subfigure
#+end_figure

** Conclusion
:PROPERTIES:
:UNNUMBERED: t
:END:

- [ ] Say that the presented control architecture in this section
  No ambition to overcome fundamental limitations of current architectures.
  Very similar to Internal Model Control [[cite:&saxena12_advan_inter_model_contr_techn]]
  Had no time to proceed to an extensive literature review to find similar control architecture and to compare them.
  Whether the propose architecture has advantages compared to already excising architecture in the literature is not clear.
- [ ] The control architecture was presented for a SISO system, but can be applied to MIMO if decoupling is sufficient.
  It will be experimentally demonstrated with the NASS.
- [ ] 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?
- [ ] Disadvantages:
  - not optimal
  - computationally intensive?
  - lead to inverse control which may not be wanted in many cases. Add reference.

* Conclusion
:PROPERTIES:
:UNNUMBERED: t
:END:
<<sec:detail_control_conclusion>>


* Bibliography                                                        :ignore:
#+latex: \printbibliography[heading=bibintoc,title={Bibliography}]

* Matlab Functions                                                  :noexport:
** =generateWF=: Generate Weighting Functions

#+begin_src matlab :tangle matlab/src/generateWF.m :comments none :mkdirp yes :eval no
function [W] = generateWF(args)
% generateWF -
%
% Syntax: [W] = generateWeight(args)
%
% Inputs:
%    - n  - Weight Order (integer)
%    - G0 - Low frequency Gain
%    - G1 - High frequency Gain
%    - Gc - Gain of the weight at frequency w0
%    - w0 - Frequency at which |W(j w0)| = Gc [rad/s]
%
% Outputs:
%    - W - Generated Weighting Function

%% Argument validation
arguments
    args.n    (1,1) double {mustBeInteger, mustBePositive} = 1
    args.G0   (1,1) double {mustBeNumeric, mustBePositive} = 0.1
    args.Ginf (1,1) double {mustBeNumeric, mustBePositive} = 10
    args.Gc   (1,1) double {mustBeNumeric, mustBePositive} = 1
    args.w0   (1,1) double {mustBeNumeric, mustBePositive} = 1
end

% Verification of correct relation between G0, Gc and Ginf
mustBeBetween(args.G0, args.Gc, args.Ginf);

%% Initialize the Laplace variable
s = zpk('s');

%% Create the weighting function according to formula
W = (((1/args.w0)*sqrt((1-(args.G0/args.Gc)^(2/args.n))/(1-(args.Gc/args.Ginf)^(2/args.n)))*s + ...
      (args.G0/args.Gc)^(1/args.n))/...
     ((1/args.Ginf)^(1/args.n)*(1/args.w0)*sqrt((1-(args.G0/args.Gc)^(2/args.n))/(1-(args.Gc/args.Ginf)^(2/args.n)))*s + ...
      (1/args.Gc)^(1/args.n)))^args.n;

%% Custom validation function
function mustBeBetween(a,b,c)
    if ~((a > b && b > c) || (c > b && b > a))
        eid = 'createWeight:inputError';
        msg = 'Gc should be between G0 and Ginf.';
        throwAsCaller(MException(eid,msg))
    end
#+end_src

** =generateCF=: Generate Complementary Filters

#+begin_src matlab :tangle matlab/src/generateCF.m :comments none :mkdirp yes :eval no
function [H1, H2] = generateCF(W1, W2, args)
% generateCF -
%
% Syntax: [H1, H2] = generateCF(W1, W2, args)
%
% Inputs:
%    - W1 - Weighting Function for H1
%    - W2 - Weighting Function for H2
%    - args:
%      - method  - H-Infinity solver ('lmi' or 'ric')
%      - display - Display synthesis results ('on' or 'off')
%
% Outputs:
%    - H1 - Generated H1 Filter
%    - H2 - Generated H2 Filter

%% Argument validation
arguments
    W1
    W2
    args.method  char {mustBeMember(args.method,{'lmi', 'ric'})} = 'ric'
    args.display char {mustBeMember(args.display,{'on', 'off'})} = 'on'
end

%% The generalized plant is defined
P = [W1 -W1;
     0   W2;
     1   0];

%% The standard H-infinity synthesis is performed
[H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', args.method, 'DISPLAY', args.display);

%% H1 is defined as the complementary of H2
H1 = 1 - H2;
#+end_src

** =plotMagUncertainty=

#+begin_src matlab :tangle matlab/src/plotMagUncertainty.m :comments none :mkdirp yes :eval no
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
#+end_src

** =plotPhaseUncertainty=

#+begin_src matlab :tangle matlab/src/plotPhaseUncertainty.m :comments none :mkdirp yes :eval no
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.unwrap    logical {mustBeNumericOrLogical} = false
        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
    if args.unwrap
        G_ang = 180/pi*unwrap(angle(squeeze(freqresp(args.G, freqs, 'Hz'))));
    else
        G_ang = 180/pi*angle(squeeze(freqresp(args.G, freqs, 'Hz')));
    end

    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
#+end_src
* Helping Functions                                                 :noexport:
** Initialize Path
#+NAME: m-init-path
#+BEGIN_SRC matlab
%% Path for functions, data and scripts
addpath('./matlab/src/'); % Path for functions
addpath('./matlab/');     % Path for scripts
#+END_SRC

#+NAME: m-init-path-tangle
#+BEGIN_SRC matlab
%% Path for functions, data and scripts
addpath('./src/'); % Path for functions
#+END_SRC

** Initialize other elements
#+NAME: m-init-other
#+BEGIN_SRC matlab
%% Colors for the figures
colors = colororder;
#+END_SRC

* Footnotes

[fn:detail_control_1]A set of two complementary filters are two transfer functions that sum to one.