Regroup all the synthesis method in one heading
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@ -820,14 +820,16 @@ We see that the blue complementary filters with a lower maximum norm permits to
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** TODO More Complete example with model uncertainty
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* Complementary filters using analytical formula
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* Methods of complementary filter synthesis
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** Complementary filters using analytical formula
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:PROPERTIES:
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:header-args:matlab+: :tangle matlab/comp_filters_analytical.m
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:header-args:matlab+: :comments org :mkdirp yes
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:END:
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<<sec:comp_filters_analytical>>
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** Introduction :ignore:
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** ZIP file containing the data and matlab files :ignore:
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*** Introduction :ignore:
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*** ZIP file containing the data and matlab files :ignore:
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#+begin_src bash :exports none :results none
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if [ matlab/comp_filters_analytical.m -nt data/comp_filters_analytical.zip ]; then
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cp matlab/comp_filters_analytical.m comp_filters_analytical.m;
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@ -841,7 +843,7 @@ We see that the blue complementary filters with a lower maximum norm permits to
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All the files (data and Matlab scripts) are accessible [[file:data/comp_filters_analytical.zip][here]].
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#+end_note
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** Matlab Init :noexport:ignore:
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*** Matlab Init :noexport:ignore:
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#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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#+end_src
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@ -854,7 +856,7 @@ We see that the blue complementary filters with a lower maximum norm permits to
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freqs = logspace(-1, 3, 1000);
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#+end_src
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** Analytical 1st order complementary filters
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*** Analytical 1st order complementary filters
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First order complementary filters are defined with following equations:
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\begin{align}
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H_L(s) = \frac{1}{1 + \frac{s}{\omega_0}}\\
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@ -906,7 +908,7 @@ Their bode plot is shown figure [[fig:comp_filter_1st_order]].
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#+CAPTION: Bode plot of first order complementary filter ([[./figs/comp_filter_1st_order.png][png]], [[./figs/comp_filter_1st_order.pdf][pdf]])
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[[file:figs/comp_filter_1st_order.png]]
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** Second Order Complementary Filters
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*** Second Order Complementary Filters
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We here use analytical formula for the complementary filters $H_L$ and $H_H$.
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The first two formulas that are used to generate complementary filters are:
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@ -1006,7 +1008,7 @@ We now study the maximum norm of the filters function of the parameter $\alpha$.
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#+CAPTION: Evolution of the H-Infinity norm of the complementary filters with the parameter $\alpha$ ([[./figs/param_alpha_hinf_norm.png][png]], [[./figs/param_alpha_hinf_norm.pdf][pdf]])
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[[file:figs/param_alpha_hinf_norm.png]]
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** Third Order Complementary Filters
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*** Third Order Complementary Filters
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The following formula gives complementary filters with slopes of $-3$ and $3$:
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\begin{align*}
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H_L(s) &= \frac{\left(1+(\alpha+1)(\beta+1)\right) (\frac{s}{\omega_0})^2 + (1+\alpha+\beta)(\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) \left( (\frac{s}{\omega_0})^2 + \beta (\frac{s}{\omega_0}) + 1 \right)}\\
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@ -1053,15 +1055,15 @@ The filters are defined below and the result is shown on figure [[fig:complement
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#+CAPTION: Third order complementary filters using the analytical formula ([[./figs/complementary_filters_third_order.png][png]], [[./figs/complementary_filters_third_order.pdf][pdf]])
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[[file:figs/complementary_filters_third_order.png]]
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* H-Infinity synthesis of complementary filters
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** H-Infinity synthesis of complementary filters
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:PROPERTIES:
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:header-args:matlab+: :tangle matlab/h_inf_synthesis_complementary_filters.m
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:header-args:matlab+: :comments org :mkdirp yes
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:END:
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<<sec:h_inf_synthesis_complementary_filters>>
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** Introduction :ignore:
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** ZIP file containing the data and matlab files :ignore:
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*** Introduction :ignore:
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*** ZIP file containing the data and matlab files :ignore:
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#+begin_src bash :exports none :results none
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if [ matlab/h_inf_synthesis_complementary_filters.m -nt data/h_inf_synthesis_complementary_filters.zip ]; then
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cp matlab/h_inf_synthesis_complementary_filters.m h_inf_synthesis_complementary_filters.m;
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@ -1075,7 +1077,7 @@ The filters are defined below and the result is shown on figure [[fig:complement
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All the files (data and Matlab scripts) are accessible [[file:data/h_inf_synthesis_complementary_filters.zip][here]].
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#+end_note
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** Matlab Init :noexport:ignore:
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*** Matlab Init :noexport:ignore:
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#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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#+end_src
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@ -1088,7 +1090,7 @@ The filters are defined below and the result is shown on figure [[fig:complement
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freqs = logspace(-1, 3, 1000);
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#+end_src
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** Synthesis Architecture
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*** Synthesis Architecture
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We here synthesize the complementary filters using the $\mathcal{H}_\infty$ synthesis.
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The goal is to specify upper bounds on the norms of $H_L$ and $H_H$ while ensuring their complementary property ($H_L + H_H = 1$).
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@ -1115,7 +1117,7 @@ We then see that $w_L$ and $w_H$ can be used to shape both $H_L$ and $H_H$ while
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#+caption: $\mathcal{H}_\infty$ synthesis of the complementary filters
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[[file:figs-tikz/sf_hinf_filters_b.png]]
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** Weights
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*** Weights
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#+begin_src matlab
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omegab = 2*pi*9;
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@ -1149,7 +1151,7 @@ We then see that $w_L$ and $w_H$ can be used to shape both $H_L$ and $H_H$ while
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#+CAPTION: Weights on the complementary filters $w_L$ and $w_H$ and the associated performance weights ([[./figs/weights_wl_wh.png][png]], [[./figs/weights_wl_wh.pdf][pdf]])
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[[file:figs/weights_wl_wh.png]]
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** H-Infinity Synthesis
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*** H-Infinity Synthesis
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We define the generalized plant $P$ on matlab.
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#+begin_src matlab
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P = [0 wL;
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@ -1189,7 +1191,7 @@ We then define the high pass filter $H_H = 1 - H_L$. The bode plot of both $H_L$
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Hh_hinf = 1 - Hl_hinf;
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#+end_src
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** Obtained Complementary Filters
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*** Obtained Complementary Filters
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The obtained complementary filters are shown on figure [[fig:hinf_filters_results]].
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@ -1223,19 +1225,19 @@ The obtained complementary filters are shown on figure [[fig:hinf_filters_result
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#+CAPTION: Obtained complementary filters using $\mathcal{H}_\infty$ synthesis ([[./figs/hinf_filters_results.png][png]], [[./figs/hinf_filters_results.pdf][pdf]])
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[[file:figs/hinf_filters_results.png]]
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* Feedback Control Architecture to generate Complementary Filters
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** Feedback Control Architecture to generate Complementary Filters
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:PROPERTIES:
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:header-args:matlab+: :tangle matlab/feedback_generate_comp_filters.m
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:header-args:matlab+: :comments org :mkdirp yes
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:END:
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<<sec:feedback_generate_comp_filters>>
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** Introduction :ignore:
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*** Introduction :ignore:
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The idea is here to use the fact that in a classical feedback architecture, $S + T = 1$, in order to design complementary filters.
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Thus, all the tools that has been developed for classical feedback control can be used for complementary filter design.
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** ZIP file containing the data and matlab files :ignore:
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*** ZIP file containing the data and matlab files :ignore:
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#+begin_src bash :exports none :results none
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if [ matlab/feedback_generate_comp_filters.m -nt data/feedback_generate_comp_filters.zip ]; then
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cp matlab/feedback_generate_comp_filters.m feedback_generate_comp_filters.m;
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@ -1249,7 +1251,7 @@ Thus, all the tools that has been developed for classical feedback control can b
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All the files (data and Matlab scripts) are accessible [[file:data/feedback_generate_comp_filters.zip][here]].
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#+end_note
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** Matlab Init :noexport:ignore:
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*** Matlab Init :noexport:ignore:
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#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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#+end_src
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@ -1262,7 +1264,7 @@ Thus, all the tools that has been developed for classical feedback control can b
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freqs = logspace(-2, 2, 1000);
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#+end_src
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** Architecture
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*** Architecture
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#+name: fig:complementary_filters_feedback_architecture
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#+caption: Architecture used to generate the complementary filters
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[[file:figs-tikz/complementary_filters_feedback_architecture.png]]
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@ -1278,7 +1280,7 @@ A simple choice is:
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Which contains two integrator and a lead. $\omega_c$ is used to tune the crossover frequency and $\alpha$ the trade-off "bump" around blending frequency and filtering away from blending frequency.
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** Loop Gain Design
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*** Loop Gain Design
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Let's first define the loop gain $L$.
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#+begin_src matlab
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wc = 2*pi*1;
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@ -1317,7 +1319,7 @@ Let's first define the loop gain $L$.
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#+CAPTION: Bode plot of the loop gain $L$ ([[./figs/loop_gain_bode_plot.png][png]], [[./figs/loop_gain_bode_plot.pdf][pdf]])
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[[file:figs/loop_gain_bode_plot.png]]
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** Complementary Filters Obtained
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*** Complementary Filters Obtained
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We then compute the resulting low pass and high pass filters.
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#+begin_src matlab
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Hl = L/(L + 1);
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@ -1353,10 +1355,10 @@ We then compute the resulting low pass and high pass filters.
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#+CAPTION: Low pass and High pass filters $H_L$ and $H_H$ for different values of $\alpha$ ([[./figs/low_pass_high_pass_filters.png][png]], [[./figs/low_pass_high_pass_filters.pdf][pdf]])
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[[file:figs/low_pass_high_pass_filters.png]]
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* Analytical Formula found in the literature
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** Analytical Formula found in the literature
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<<sec:analytical_formula_literature>>
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** Analytical Formula
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*** Analytical Formula
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cite:min15_compl_filter_desig_angle_estim
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\begin{align*}
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H_L(s) = \frac{K_p s + K_i}{s^2 + K_p s + K_i} \\
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@ -1392,7 +1394,7 @@ cite:baerveldt97_low_cost_low_weigh_attit
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H_H(s) = \frac{\tau^2 s^2}{(\tau s + 1)^2}
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\end{align*}
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** Matlab Init :noexport:ignore:
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*** Matlab Init :noexport:ignore:
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#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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#+end_src
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@ -1401,7 +1403,7 @@ cite:baerveldt97_low_cost_low_weigh_attit
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<<matlab-init>>
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#+end_src
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** Matlab
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*** Matlab
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#+begin_src matlab
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omega0 = 1*2*pi; % [rad/s]
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tau = 1/omega0; % [s]
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@ -1461,10 +1463,10 @@ cite:baerveldt97_low_cost_low_weigh_attit
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#+CAPTION: Comparison of some complementary filters found in the literature ([[./figs/comp_filters_literature.png][png]], [[./figs/comp_filters_literature.pdf][pdf]])
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[[file:figs/comp_filters_literature.png]]
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** Discussion
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*** Discussion
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Analytical Formula found in the literature provides either no parameter for tuning the robustness / performance trade-off.
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* Comparison of the different methods of synthesis
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** Comparison of the different methods of synthesis
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<<sec:discussion>>
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The generated complementary filters using $\mathcal{H}_\infty$ and the analytical formulas are very close to each other. However there is some difference to note here:
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- the analytical formula provides a very simple way to generate the complementary filters (and thus the controller), they could even be used to tune the controller online using the parameters $\alpha$ and $\omega_0$. However, these formula have the property that $|H_H|$ and $|H_L|$ are symmetrical with the frequency $\omega_0$ which may not be desirable.
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