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index.org
32
index.org
@ -17,7 +17,7 @@
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#+BIND: org-latex-image-default-width ""
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#+LaTeX_CLASS: scrreprt
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#+LaTeX_CLASS_OPTIONS: [a4paper, 10pt, DIV=12]
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#+LaTeX_CLASS_OPTIONS: [a4paper, 10pt, DIV=12, parskip=full]
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#+LaTeX_HEADER_EXTRA: \input{preamble.tex}
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#+CSL_STYLE: ieee.csl
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@ -50,7 +50,7 @@
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* LaTeX Config :noexport:
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#+begin_src emacs-lisp :tangle no
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(setq org-latex-default-table-environment "tabular")
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(setq org-latex-tables-booktabs nil)
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(setq org-latex-tables-booktabs t)
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#+end_src
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#+begin_src latex :tangle preamble.tex
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@ -210,7 +210,6 @@ Therefore, for more advanced discussion, please have a look at the recommended r
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When possible, Matlab scripts used for the example/exercises are provided such that you can easily test them on your computer.
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The general structure of this document is as follows:
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- A short introduction to /model based control/ is given in Section [[sec:model_based_control]]
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- Classical /open/ loop shaping method is presented in Section [[sec:open_loop_shaping]].
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@ -316,7 +315,7 @@ Note that in parallel, there have been numerous other developments, including no
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#+name: tab:comparison_control_methods
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#+caption: Table summurazing the main differences between classical, modern and robust control
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#+attr_latex: :environment tabularx :booktabs t :width \linewidth :align lccc
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#+attr_latex: :environment tabularx :width \linewidth :align lccc
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| <l> | <c> | <c> | <c> |
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| | *Classical Control* | *Modern Control* | *Robust Control* |
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|---------------+---------------------+--------------------------+-------------------------------|
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@ -468,17 +467,19 @@ The notations used on Figure [[fig:mech_sys_1dof_inertial_contr]] are listed and
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#+name: tab:example_notations
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#+caption: Example system variables
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#+attr_latex: :environment tabularx :booktabs t :width \linewidth :align cXcc
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#+attr_latex: :environment tabularx :width \linewidth :align cXcc
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| *Notation* | *Description* | *Value* | *Unit* |
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|--------------------+----------------------------------------------------------------+----------------+-----------|
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| $m$ | Payload's mass to position / isolate | $10$ | [kg] |
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| $k$ | Stiffness of the suspension system | $10^6$ | [N/m] |
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| $c$ | Damping coefficient of the suspension system | $400$ | [N/(m/s)] |
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|--------------------+----------------------------------------------------------------+----------------+-----------|
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| $y$ | Payload absolute displacement (measured by an inertial sensor) | | [m] |
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| $d$ | Ground displacement, it acts as a disturbance | | [m] |
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| $u$ | Actuator force | | [N] |
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| $r$ | Wanted position of the mass (the reference) | | [m] |
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| $\epsilon = r - y$ | Position error | | [m] |
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|--------------------+----------------------------------------------------------------+----------------+-----------|
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| $K$ | Feedback controller | to be designed | [N/m] |
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#+begin_exercice
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@ -563,12 +564,13 @@ The Bode plots of $G(s)$ and $G_d(s)$ are shown in Figures [[fig:bode_plot_examp
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
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hold off;
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ylim([1e-9, 1e-5]);
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ax2 = nexttile;
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hold on;
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G, freqs, 'Hz')))));
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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yticks(-360:90:360); ylim([-270, 90]);
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yticks(-360:90:360); ylim([-200, 20]);
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xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
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hold off;
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linkaxes([ax1,ax2],'x');
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@ -576,7 +578,7 @@ The Bode plots of $G(s)$ and $G_d(s)$ are shown in Figures [[fig:bode_plot_examp
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#+end_src
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#+begin_src matlab :tangle no :exports results :results file replace
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exportFig('figs/bode_plot_example_afm.pdf', 'width', 'wide', 'height', 'normal');
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exportFig('figs/bode_plot_example_afm.pdf', 'width', 'wide', 'height', 'tall');
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#+end_src
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#+name: fig:bode_plot_example_afm
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@ -845,7 +847,7 @@ Such shape corresponds to the typical wanted Loop gain Shape shown in Figure [[f
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#+name: tab:open_loop_shaping_specifications
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#+caption: Wanted Loop Shape corresponding to each specification
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#+attr_latex: :environment tabularx :booktabs t :width \linewidth :align lXX
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#+attr_latex: :environment tabularx :width \linewidth :align lXX
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| | *Specification* | *Corresponding Loop Shape* |
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|--------------+----------------------------------------+------------------------------------------|
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| *Dist. Rej.* | Highest possible rejection below 1Hz | Slope of -40dB/dec at low frequency |
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@ -967,7 +969,7 @@ Let's finally compare the obtained stability margins of the $\mathcal{H}_\infty$
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#+name: tab:open_loop_shaping_compare
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#+caption: Comparison of the characteristics obtained with the two methods
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#+attr_latex: :environment tabularx :booktabs t :width \linewidth :align Xcc
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#+attr_latex: :environment tabularx :width \linewidth :align Xcc
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#+RESULTS:
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| *Specifications* | *Manual Method* | *$\mathcal{H}_\infty$ Method* |
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|-----------------------------+-----------------+-------------------------------|
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@ -1108,7 +1110,7 @@ A practical example about how to derive the generalized plant for a classical co
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#+name: tab:notation_general
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#+caption: Notations for the general configuration
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#+attr_latex: :environment tabularx :booktabs t :width 0.8\linewidth :align cX :float nil
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#+attr_latex: :environment tabularx :width 0.8\linewidth :align cX :float nil
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| *Notation* | *Meaning* |
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|------------+----------------------------------------------------|
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| $P$ | Generalized plant model |
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@ -1329,7 +1331,7 @@ These are summarized in Table [[tab:spec_closed_loop_tf]].
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#+name: tab:spec_closed_loop_tf
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#+caption: Typical Specification and associated closed-loop transfer function
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#+attr_latex: :environment tabularx :booktabs t :width 0.8\linewidth :align Xl
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#+attr_latex: :environment tabularx :width 0.8\linewidth :align Xl
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| *Specification* | *CL Transfer Function* |
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|--------------------------------+-----------------------------------------------|
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| Reference Tracking | From $r$ to $\epsilon$ |
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@ -1409,7 +1411,7 @@ The comparison of the sensitivity functions shapes and their effect on the step
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#+name: tab:compare_sensitivity_shapes
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#+caption: Comparison of the sensitivity function shape and the corresponding step response for the three controller variations
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#+attr_latex: :environment tabularx :booktabs t :width 0.9\linewidth :align lXX
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#+attr_latex: :environment tabularx :width 0.9\linewidth :align lXX
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| *Controller* | *Sensitivity Function Shape* | *Change of the Step Response* |
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|--------------+----------------------------------------------------+----------------------------------|
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| $K_1(s)$ | Larger bandwidth $\omega_b$ | Faster rise time |
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@ -1817,7 +1819,7 @@ And we now understand why setting an upper bound on the magnitude of $S$ is gene
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#+name: tab:specification_modern
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#+caption: Typical Specifications and corresponding wanted norms of open and closed loop tansfer functions
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#+attr_latex: :environment tabularx :booktabs t :width 0.9\linewidth :align lXX
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#+attr_latex: :environment tabularx :width 0.9\linewidth :align lXX
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| | *Open-Loop Shaping* | *Closed-Loop Shaping* |
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|-----------------------------+---------------------+--------------------------------------------|
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| Reference Tracking | $L$ large | $S$ small |
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@ -2244,7 +2246,7 @@ When multiple closed-loop transfer function are shaped at the same time, it is r
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#+name: tab:usual_shaping_gang_four
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#+caption: Typical specifications and corresponding shaping of the /Gang of four/
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#+attr_latex: :environment tabularx :booktabs t :width 0.9\linewidth :align llX
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#+attr_latex: :environment tabularx :width 0.9\linewidth :align llX
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| <l> | <c> | <l> |
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| *Specifications* | *TF* | *Wanted shape* |
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|-------------------------------+------+------------------------------------------------|
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@ -3029,7 +3031,7 @@ In such case, we want to shape $S$, $GS$ and $T$.
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#+name: tab:ex_specification_shapes
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#+caption: Control Specifications and associated wanted shape of the closed-loop transfer functions
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#+attr_latex: :environment tabularx :booktabs t :width 0.7\linewidth :align llX
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#+attr_latex: :environment tabularx :width 0.7\linewidth :align llX
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| <l> | <c> | <l> |
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||||
| *Specification* | *TF* | *Wanted Shape* |
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||||
|--------------------------+-----------+------------------------------------|
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80
index.tex
80
index.tex
@ -1,6 +1,6 @@
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% Created 2020-12-04 ven. 23:35
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% Created 2020-12-04 ven. 23:52
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% Intended LaTeX compiler: pdflatex
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\documentclass[a4paper, 10pt, DIV=12]{scrreprt}
|
||||
\documentclass[a4paper, 10pt, DIV=12, parskip=full]{scrreprt}
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||||
\usepackage[utf8]{inputenc}
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\usepackage[T1]{fontenc}
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\usepackage{graphicx}
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@ -38,7 +38,7 @@
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\chapter*{Introduction}
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\label{sec:orge367402}
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\label{sec:orgd1c4555}
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The purpose of this document is to give a \emph{practical introduction} to the wonderful world of \(\mathcal{H}_\infty\) Control.
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No attend is made to provide an exhaustive treatment of the subject.
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@ -61,11 +61,11 @@ Such technique is presented in Section \ref{sec:closed-loop-shaping}
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\end{itemize}
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\chapter{Introduction to Model Based Control}
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\label{sec:orgef98240}
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||||
\label{sec:org906ea4a}
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\label{sec:model_based_control}
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\section{Model Based Control - Methodology}
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\label{sec:orge1fd1b8}
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||||
\label{sec:org56271e2}
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\label{sec:model_based_control_methodology}
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||||
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The typical methodology for \textbf{Model Based Control} techniques is schematically shown in Figure \ref{fig:control-procedure}.
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@ -95,7 +95,7 @@ In Section \ref{sec:open_loop_shaping}, steps 2 and 3 will be described for a co
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Then, steps 2 and 3 for the \textbf{\(\mathcal{H}_\infty\) Loop Shaping} of closed-loop transfer functions will be discussed in Sections \ref{sec:modern_interpretation_specification}, \ref{sec:closed-loop-shaping} and \ref{sec:h_infinity_mixed_sensitivity}.
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\section{From Classical Control to Robust Control}
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||||
\label{sec:org80f6e15}
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||||
\label{sec:orgd4cb731}
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||||
\label{sec:comp_classical_modern_robust_control}
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||||
Many different model based control techniques have been developed since the birth of \emph{classical control theory} in the '30s.
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@ -154,7 +154,7 @@ Note that in parallel, there have been numerous other developments, including no
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\end{table}
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\section{Example System}
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||||
\label{sec:orgd6f2ecd}
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\label{sec:org24dcae9}
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||||
\label{sec:example_system}
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||||
Throughout this document, multiple examples and practical application of presented control strategies will be provided.
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@ -254,7 +254,7 @@ The Bode plots of \(G(s)\) and \(G_d(s)\) are shown in Figures \ref{fig:bode_plo
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\end{figure}
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\chapter{Classical Open Loop Shaping}
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\label{sec:org88c26ac}
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\label{sec:org2383b8a}
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\label{sec:open_loop_shaping}
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After an introduction to classical Loop Shaping in Section \ref{sec:open_loop_shaping_introduction}, a practical example is given in Section \ref{sec:loop_shaping_example}.
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Such Loop Shaping is usually performed manually with tools coming from the classical control theory.
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@ -263,7 +263,7 @@ However, the \(\mathcal{H}_\infty\) synthesis can be used to automate the Loop S
|
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This is presented in Section \ref{sec:h_infinity_open_loop_shaping} and applied on the same example in Section \ref{sec:h_infinity_open_loop_shaping_example}.
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\section{Introduction to Loop Shaping}
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\label{sec:orge3291a7}
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\label{sec:orgca87141}
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\label{sec:open_loop_shaping_introduction}
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\begin{definition}
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@ -311,7 +311,7 @@ But this is were the \(\mathcal{H}_\infty\) Synthesis will be useful!
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More details on that in Sections \ref{sec:modern_interpretation_specification} and \ref{sec:closed-loop-shaping}.
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\section{Example of Manual Open Loop Shaping}
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\label{sec:org06d32ed}
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\label{sec:orgc736296}
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\label{sec:loop_shaping_example}
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\begin{exampl}
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@ -360,11 +360,13 @@ The bode plot of the Loop Gain is shown in Figure \ref{fig:loop_gain_manual_afm}
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\begin{center}
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\begin{tabular}{lr}
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\toprule
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Requirements & Manual Method\\
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\hline
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\midrule
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Gain Margin \(> 3\) [dB] & 3.1\\
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Phase Margin \(> 30\) [deg] & 35.4\\
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Crossover \(\approx 10\) [Hz] & 10.1\\
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\bottomrule
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\end{tabular}
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\end{center}
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@ -375,7 +377,7 @@ Crossover \(\approx 10\) [Hz] & 10.1\\
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\end{figure}
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\section{\(\mathcal{H}_\infty\) Loop Shaping Synthesis}
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\label{sec:org2bb176d}
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\label{sec:orga9060d3}
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\label{sec:h_infinity_open_loop_shaping}
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The synthesis of controllers based on the Loop Shaping method can be automated using the \(\mathcal{H}_\infty\) Synthesis.
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@ -400,7 +402,7 @@ Therefore, by just providing the wanted loop shape and the plant model, the \(\m
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Even though we will not go into details and explain how such synthesis is working, an example is provided in the next section.
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\section{Example of the \(\mathcal{H}_\infty\) Loop Shaping Synthesis}
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\label{sec:org4ea69be}
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\label{sec:orgc755cbe}
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\label{sec:h_infinity_open_loop_shaping_example}
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To apply the \(\mathcal{H}_\infty\) Loop Shaping Synthesis, the wanted shape of the loop gain should be determined from the specifications.
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@ -485,7 +487,7 @@ Crossover \(\approx 10\) [Hz] & 10.1 & 9.9\\
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\end{table}
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\chapter{A first Step into the \(\mathcal{H}_\infty\) world}
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\label{sec:org9957ebe}
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\label{sec:orgfc80984}
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\label{sec:h_infinity_introduction}
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In this section, the \(\mathcal{H}_\infty\) Synthesis method, which is based on the optimization of the \(\mathcal{H}_\infty\) norm of transfer functions, is introduced.
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@ -497,7 +499,7 @@ The \(\mathcal{H}_\infty\) is then applied to this generalized plant in Section
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Finally, an example showing how to convert a typical feedback control architecture into a generalized plant is given in Section \ref{sec:generalized_plant_derivation}.
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\section{The \(\mathcal{H}_\infty\) Norm}
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\label{sec:org4d1191c}
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\label{sec:org23de71d}
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\label{sec:h_infinity_norm}
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\begin{definition}
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@ -532,7 +534,7 @@ We can see in Figure \ref{fig:hinfinity_norm_siso_bode} that indeed, the \(\math
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\end{exampl}
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\section{\(\mathcal{H}_\infty\) Synthesis}
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\label{sec:org25f160e}
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\label{sec:org251bca8}
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\label{sec:h_infinity_synthesis}
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\begin{definition}
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@ -562,7 +564,7 @@ Note that there are many ways to use the \(\mathcal{H}_\infty\) Synthesis:
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\end{itemize}
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\section{The Generalized Plant}
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\label{sec:org1d5b54d}
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\label{sec:orgada260f}
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\label{sec:generalized_plant}
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The first step when applying the \(\mathcal{H}_\infty\) synthesis is usually to write the problem as a standard \(\mathcal{H}_\infty\) problem.
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@ -601,7 +603,7 @@ A practical example about how to derive the generalized plant for a classical co
|
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\end{important}
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\section{The \(\mathcal{H}_\infty\) Synthesis applied on the Generalized plant}
|
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\label{sec:orgd38e75b}
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||||
\label{sec:org1d8fc5f}
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\label{sec:h_infinity_general_synthesis}
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Once the generalized plant is obtained, the \(\mathcal{H}_\infty\) synthesis problem can be stated as follows:
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@ -637,7 +639,7 @@ where:
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Note that the general control configure of Figure \ref{fig:general_control_names}, as its name implies, is quite \emph{general} and can represent feedback control as well as feedforward control architectures.
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\section{From a Classical Feedback Architecture to a Generalized Plant}
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\label{sec:orgd8a0711}
|
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\label{sec:orgead7d16}
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\label{sec:generalized_plant_derivation}
|
||||
|
||||
The procedure to convert a typical control architecture as the one shown in Figure \ref{fig:classical_feedback_tracking} to a generalized Plant is as follows:
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@ -689,7 +691,7 @@ P.OutputName = {'e', 'u', 'v'};
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\end{exercice}
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\chapter{Modern Interpretation of Control Specifications}
|
||||
\label{sec:orgcd910fe}
|
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\label{sec:orgf4eb6c5}
|
||||
\label{sec:modern_interpretation_specification}
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||||
As shown in Section \ref{sec:open_loop_shaping}, the loop gain \(L(s) = G(s) K(s)\) is a useful and easy tool when manually designing controllers.
|
||||
This is mainly due to the fact that \(L(s)\) is very easy to shape as it depends \emph{linearly} on \(K(s)\).
|
||||
@ -713,7 +715,7 @@ The robustness (stability margins) of the system can also be linked to the shape
|
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Links between typical control specifications and shapes of the closed-loop transfer functions are summarized in Section \ref{sec:other_requirements}.
|
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|
||||
\section{Closed Loop Transfer Functions and the Gang of Four}
|
||||
\label{sec:org6d77210}
|
||||
\label{sec:org40d15a2}
|
||||
\label{sec:closed_loop_tf}
|
||||
|
||||
Consider the typical feedback system shown in Figure \ref{fig:gang_of_four_feedback}.
|
||||
@ -798,7 +800,7 @@ Thus, for reference tracking, we have to shape the \emph{closed-loop} transfer f
|
||||
Similarly, to reduce the effect of measurement noise \(n\) on the output \(y\), we have to act on the complementary sensitivity function \(T(s)\).
|
||||
|
||||
\section{The Sensitivity Function}
|
||||
\label{sec:org2c7250a}
|
||||
\label{sec:org6e8bfcd}
|
||||
\label{sec:sensitivity_transfer_functions}
|
||||
|
||||
The sensitivity function is indisputably the most important closed-loop transfer function of a feedback system.
|
||||
@ -861,7 +863,7 @@ This will become clear in the next section about the \textbf{module margin}.
|
||||
\end{important}
|
||||
|
||||
\section{Robustness: Module Margin}
|
||||
\label{sec:org30ae1f0}
|
||||
\label{sec:org47b850d}
|
||||
\label{sec:module_margin}
|
||||
|
||||
Let's start this section by an example demonstrating why the phase and gain margins might not be good indicators of robustness.
|
||||
@ -1002,7 +1004,7 @@ To learn more about module/disk margin, you can check out \href{https://www.yout
|
||||
\end{seealso}
|
||||
|
||||
\section{Summary of typical specification and associated wanted shaping}
|
||||
\label{sec:org3eb7db2}
|
||||
\label{sec:org5f5ba8c}
|
||||
\label{sec:other_requirements}
|
||||
|
||||
\begin{table}[htbp]
|
||||
@ -1022,7 +1024,7 @@ Robustness & Phase/Gain margins & Module margin: \(\Vert S\Vert_\infty\) small\\
|
||||
\end{table}
|
||||
|
||||
\chapter{\(\mathcal{H}_\infty\) Shaping of closed-loop transfer functions}
|
||||
\label{sec:org80e0308}
|
||||
\label{sec:org2b02046}
|
||||
\label{sec:closed-loop-shaping}
|
||||
In the previous sections, we have seen that the performances of the system depends on the \textbf{shape} of the closed-loop transfer function.
|
||||
Therefore, the synthesis problem is to design \(K(s)\) such that closed-loop system is stable and such that the closed-loop transfer functions such as \(S\), \(KS\) and \(T\) are shaped as wanted.
|
||||
@ -1041,7 +1043,7 @@ Such synthesis is usually called \textbf{Mixed-sensitivity Loop Shaping} and is
|
||||
Some insight on the use and limitations of such techniques are given in Section \ref{sec:shaping_multiple_tf}.
|
||||
|
||||
\section{How to Shape closed-loop transfer function? Using Weighting Functions!}
|
||||
\label{sec:orga64410a}
|
||||
\label{sec:org6ed8543}
|
||||
\label{sec:weighting_functions}
|
||||
|
||||
Suppose we apply the \(\mathcal{H}_\infty\) synthesis on the generalized plant \(P(s)\) shown in Figure \ref{fig:loop_shaping_S_without_W}.
|
||||
@ -1110,7 +1112,7 @@ Pw = blkdiag(Ws, 1)*P;
|
||||
\end{exercice}
|
||||
|
||||
\section{Design of Weighting Functions}
|
||||
\label{sec:org0c721f2}
|
||||
\label{sec:org6cb9a44}
|
||||
\label{sec:weighting_functions_design}
|
||||
|
||||
Weighting function included in the generalized plant must be \textbf{proper}, \textbf{stable} and \textbf{minimum phase} transfer functions.
|
||||
@ -1217,7 +1219,7 @@ The obtained shapes are shown in Figure \ref{fig:high_order_weight}.
|
||||
\end{seealso}
|
||||
|
||||
\section{Shaping the Sensitivity Function}
|
||||
\label{sec:org622981c}
|
||||
\label{sec:org9084e2e}
|
||||
\label{sec:sensitivity_shaping_example}
|
||||
|
||||
Let's design a controller using the \(\mathcal{H}_\infty\) shaping of the sensitivity function that fulfils the following requirements:
|
||||
@ -1325,7 +1327,7 @@ It just means that at some frequency, one of the closed-loop transfer functions
|
||||
\end{figure}
|
||||
|
||||
\section{Shaping multiple closed-loop transfer functions - Limitations}
|
||||
\label{sec:org501a0b0}
|
||||
\label{sec:org08a5a05}
|
||||
\label{sec:shaping_multiple_tf}
|
||||
As was shown in Section \ref{sec:modern_interpretation_specification}, each of the four main closed-loop transfer functions (called the \emph{gang of four}) will impact different characteristics of the closed-loop system.
|
||||
This is summarized in Table \ref{tab:usual_shaping_gang_four}.
|
||||
@ -1557,7 +1559,7 @@ Two approaches can be used to obtain controllers with reasonable order:
|
||||
\end{warning}
|
||||
|
||||
\chapter{Mixed-Sensitivity \(\mathcal{H}_\infty\) Control - Example}
|
||||
\label{sec:orgcbf596f}
|
||||
\label{sec:org4999065}
|
||||
\label{sec:h_infinity_mixed_sensitivity}
|
||||
Let's now apply the \(\mathcal{H}_\infty\) Shaping control procedure on a practical example.
|
||||
|
||||
@ -1569,7 +1571,7 @@ The important step of interpreting the specifications as wanted shape of closed-
|
||||
Finally, the shaping of closed-loop transfer functions is performed in Sections \ref{sec:ex_shaping_S}, \ref{sec:ex_shaping_GS} and \ref{sec:ex_shaping_T}.
|
||||
|
||||
\section{Control Problem}
|
||||
\label{sec:org0d0e2a9}
|
||||
\label{sec:org4641813}
|
||||
\label{sec:ex_control_problem}
|
||||
|
||||
Let's consider our usual \emph{test system} shown in Figure \ref{fig:ex_test_system}.
|
||||
@ -1598,7 +1600,7 @@ The considered inputs are:
|
||||
\end{itemize}
|
||||
|
||||
\section{Control Design Procedure}
|
||||
\label{sec:org0f986e4}
|
||||
\label{sec:org2f6babe}
|
||||
\label{sec:ex_control_procedure}
|
||||
|
||||
Here is the general design procedure that will be followed:
|
||||
@ -1688,7 +1690,7 @@ u = z(:,2); % Input usage [N]
|
||||
\end{minted}
|
||||
|
||||
\section{Modern Interpretation of control specifications}
|
||||
\label{sec:org1c39f44}
|
||||
\label{sec:orga74724d}
|
||||
\label{sec:ex_specification_interpretation}
|
||||
|
||||
\begin{exercice}
|
||||
@ -1769,7 +1771,7 @@ Now let's shape the three closed-loop transfer functions sequentially:
|
||||
\end{itemize}
|
||||
|
||||
\section{Step 1 - Shaping of \(S\)}
|
||||
\label{sec:org2b353b4}
|
||||
\label{sec:orgfba6439}
|
||||
\label{sec:ex_shaping_S}
|
||||
|
||||
Let's first shape the Sensitivity function as it is usually the most important of the \emph{Gang of four} closed-loop transfer functions.
|
||||
@ -1861,7 +1863,7 @@ The time domain signals are shown in Figure \ref{fig:ex_time_domain_1b} and it i
|
||||
\end{figure}
|
||||
|
||||
\section{Step 2 - Shaping of \(GS\)}
|
||||
\label{sec:org040835b}
|
||||
\label{sec:org0c2dbb8}
|
||||
\label{sec:ex_shaping_GS}
|
||||
|
||||
Looking at Figure \ref{fig:ex_results_2}, it is clear that the rejection of disturbances is not satisfactory.
|
||||
@ -1893,7 +1895,7 @@ If is shown that indeed, the disturbance rejection performance are much better a
|
||||
\end{figure}
|
||||
|
||||
\section{Step 3 - Shaping of \(T\)}
|
||||
\label{sec:orge27b967}
|
||||
\label{sec:org98a0d1b}
|
||||
\label{sec:ex_shaping_T}
|
||||
|
||||
Finally, we want to limit the effect of the noise on the displacement output.
|
||||
@ -1942,7 +1944,7 @@ This can be seen when zooming on the output signal in Figure \ref{fig:ex_time_do
|
||||
\end{figure}
|
||||
|
||||
\section{Conclusion and Discussion}
|
||||
\label{sec:org8ee7c81}
|
||||
\label{sec:org544ddf1}
|
||||
|
||||
Hopefully this practical example will help you apply the \(\mathcal{H}_\infty\) Shaping synthesis on other control problems.
|
||||
|
||||
@ -1951,7 +1953,7 @@ As an exercise, plot and analyze the evolution of the controller and loop gain t
|
||||
If the large input usage is considered to be not acceptable, the shaping of \(KS\) could be included in the synthesis and all the \emph{Gang of four} closed-loop transfer function shapes.
|
||||
|
||||
\chapter{Conclusion}
|
||||
\label{sec:org6c74b5a}
|
||||
\label{sec:org6166298}
|
||||
\label{sec:conclusion}
|
||||
|
||||
Hopefully, this document gave you a glimpse on how useful and powerful the \(\mathcal{H}_\infty\) loop shaping synthesis can be.
|
||||
@ -1959,7 +1961,7 @@ One of the true power of \(\mathcal{H}_\infty\) synthesis is that is can easily
|
||||
If you want to know more about the ``\(\mathcal{H}_\infty\) and robust control world'' some resources are given below.
|
||||
|
||||
\chapter*{Resources}
|
||||
\label{sec:org17d3722}
|
||||
\label{sec:orgd57bda1}
|
||||
For a complete treatment of multivariable robust control, I would highly recommend this book \cite{skogestad07_multiv_feedb_contr}.
|
||||
If you want to nice reference book in French, look at \cite{duc99_comman_h}.
|
||||
|
||||
|
||||
Loading…
Reference in New Issue
Block a user