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This is studied using the equations, and some numerical computations are used to compare the use of voice coil and piezoelectric actuators. +In sections [[sec:iff]] and [[sec:dvf]], the use of Integral Force Feedback and Direct Velocity Feedback is studied for the case of rotating positioning platforms. + Then, in section [[sec:control_strategies]], two different control approach are compared where: - the measurement is made in the fixed frame - the measurement is made in the rotating frame In section [[sec:simscape]], the analytical study will be validated using a multi body model of the studied system. -Finally, in section [[sec:control]], the control strategies are implemented using Simulink and Simscape and compared. - * System Description and Analysis :PROPERTIES: :HEADER-ARGS:matlab+: :tangle matlab/system_numerical_analysis.m @@ -1236,7 +1236,7 @@ As shown by the Root Locus in Figure [[fig:dvf_root_locus_ws]]: * Control Strategies - <> +<> ** Measurement in the fixed reference frame First, let's consider a measurement in the fixed referenced frame. @@ -1270,10 +1270,10 @@ The corresponding block diagram is shown figure [[fig:control_measure_rotating_2 The loop gain is $L = G K$. * Multi Body Model - Simscape - :PROPERTIES: - :HEADER-ARGS:matlab+: :tangle matlab/simscape_analysis.m - :END: - <> +:PROPERTIES: +:HEADER-ARGS:matlab+: :tangle matlab/simscape_analysis.m +:END: +<> ** Matlab Init :noexport:ignore: #+begin_src matlab :tangle no :exports none :noweb yes :var current_dir=(file-name-directory buffer-file-name) @@ -1998,47 +1998,3 @@ First, we create the closed loop systems. Then, we plot the transfer function fr The close-loop performance does not vary a lot with the rotating speed (figure [[fig:perfcomp]]) even tough the open loop system is varying quite a lot (figure [[fig:Guu_uv_ws]]). #+end_important -** BKMK Plant Control - MIMO approach -*** Analysis - SVD -The singular value decomposition of a MIMO system $G$ is defined as follow: -\[ G = U \Sigma V^H \] - -With: -- $\Sigma$ is an $2 \times 2$ matrix with 2 non-negative *singular values* $\sigma_i$, arranged in descending order along its main diagonal -- $U$ is an $2 \times 2$ unitary matrix. The columns vectors of $U$, denoted $u_i$, represent the *output directions* of the plant. They are orthonomal -- $V$ is an $2 \times 2$ unitary matrix. The columns vectors of $V$, denoted $v_i$, represent the *input directions* of the plant. They are orthonomal - -We first look at the evolution of the singular values as a function of frequency (figure [[fig:G_sigma]]). - -#+begin_src matlab :exports none - freqs = logspace(-2, 1, 1000); - - figure; - hold on; - for i = 1:length(ws) - sv = sigma(Gs_vc{i}, 2*pi*freqs); - set(gca,'ColorOrderIndex',i) - plot(freqs, sv(1, :), 'DisplayName', sprintf('w = %.0f rpm', ws(i)*60/2/pi)); - set(gca,'ColorOrderIndex',i) - plot(freqs, sv(2, :), '--', 'HandleVisibility', 'off'); - end - hold off; - set(gca,'xscale','log'); set(gca,'yscale','log'); - legend('location', 'southwest'); -#+end_src - -#+HEADER: :tangle no :exports results :results none :noweb yes -#+begin_src matlab :var filepath="figs/G_sigma.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") - <> -#+end_src - -#+LABEL: fig:G_sigma -#+CAPTION: Evolution of the singular values with frequency -[[file:figs/G_sigma.png]] - -* Control Implementation - <> - -* Bibliography :ignore: -# bibliographystyle:unsrt -# bibliography:refs.bib diff --git a/references.html b/references.html deleted file mode 100644 index f7cd043..0000000 Binary files a/references.html and /dev/null differ diff --git a/refs.bib b/refs.bib deleted file mode 100644 index e69de29..0000000 diff --git a/rotating_frame.slxc b/rotating_frame.slxc deleted file mode 100644 index 330f6f5..0000000 Binary files a/rotating_frame.slxc and /dev/null differ diff --git a/rotating_frame_ctrl.slx b/rotating_frame_ctrl.slx deleted file mode 100644 index ee27159..0000000 Binary files a/rotating_frame_ctrl.slx and /dev/null differ diff --git a/rotating_frame_ctrl.slxc b/rotating_frame_ctrl.slxc deleted file mode 100644 index dd3a8cd..0000000 Binary files a/rotating_frame_ctrl.slxc and /dev/null differ