Export to PDF!

.gitignore

@@ -0,0 +1,262 @@
+.auctex-auto/
+.log/
+
+## Core latex/pdflatex auxiliary files:
+*.aux
+*.lof
+*.log
+*.lot
+*.fls
+*.out
+*.toc
+*.fmt
+*.fot
+*.cb
+*.cb2
+.*.lb
+
+## Intermediate documents:
+*.dvi
+*.xdv
+*-converted-to.*
+# these rules might exclude image files for figures etc.
+# *.ps
+# *.eps
+# *.pdf
+
+## Generated if empty string is given at "Please type another file name for output:"
+.pdf
+
+## Bibliography auxiliary files (bibtex/biblatex/biber):
+*.bbl
+*.bcf
+*.blg
+*-blx.aux
+*-blx.bib
+*.run.xml
+
+## Build tool auxiliary files:
+*.fdb_latexmk
+*.synctex
+*.synctex(busy)
+*.synctex.gz
+*.synctex.gz(busy)
+*.pdfsync
+
+## Build tool directories for auxiliary files
+# latexrun
+latex.out/
+
+## Auxiliary and intermediate files from other packages:
+# algorithms
+*.alg
+*.loa
+
+# achemso
+acs-*.bib
+
+# amsthm
+*.thm
+
+# beamer
+*.nav
+*.pre
+*.snm
+*.vrb
+
+# changes
+*.soc
+
+# comment
+*.cut
+
+# cprotect
+*.cpt
+
+# elsarticle (documentclass of Elsevier journals)
+*.spl
+
+# endnotes
+*.ent
+
+# fixme
+*.lox
+
+# feynmf/feynmp
+*.mf
+*.mp
+*.t[1-9]
+*.t[1-9][0-9]
+*.tfm
+
+#(r)(e)ledmac/(r)(e)ledpar
+*.end
+*.?end
+*.[1-9]
+*.[1-9][0-9]
+*.[1-9][0-9][0-9]
+*.[1-9]R
+*.[1-9][0-9]R
+*.[1-9][0-9][0-9]R
+*.eledsec[1-9]
+*.eledsec[1-9]R
+*.eledsec[1-9][0-9]
+*.eledsec[1-9][0-9]R
+*.eledsec[1-9][0-9][0-9]
+*.eledsec[1-9][0-9][0-9]R
+
+# glossaries
+*.acn
+*.acr
+*.glg
+*.glo
+*.gls
+*.glsdefs
+
+# gnuplottex
+*-gnuplottex-*
+
+# gregoriotex
+*.gaux
+*.gtex
+
+# htlatex
+*.4ct
+*.4tc
+*.idv
+*.lg
+*.trc
+*.xref
+
+# hyperref
+*.brf
+
+# knitr
+*-concordance.tex
+# TODO Comment the next line if you want to keep your tikz graphics files
+*.tikz
+*-tikzDictionary
+
+# listings
+*.lol
+
+# makeidx
+*.idx
+*.ilg
+*.ind
+*.ist
+
+# minitoc
+*.maf
+*.mlf
+*.mlt
+*.mtc[0-9]*
+*.slf[0-9]*
+*.slt[0-9]*
+*.stc[0-9]*
+
+# minted
+_minted*
+*.pyg
+
+# morewrites
+*.mw
+
+# nomencl
+*.nlg
+*.nlo
+*.nls
+
+# pax
+*.pax
+
+# pdfpcnotes
+*.pdfpc
+
+# sagetex
+*.sagetex.sage
+*.sagetex.py
+*.sagetex.scmd
+
+# scrwfile
+*.wrt
+
+# sympy
+*.sout
+*.sympy
+sympy-plots-for-*.tex/
+
+# pdfcomment
+*.upa
+*.upb
+
+# pythontex
+*.pytxcode
+pythontex-files-*/
+
+# tcolorbox
+*.listing
+
+# thmtools
+*.loe
+
+# TikZ & PGF
+*.dpth
+*.md5
+*.auxlock
+
+# todonotes
+*.tdo
+
+# vhistory
+*.hst
+*.ver
+
+# easy-todo
+*.lod
+
+# xcolor
+*.xcp
+
+# xmpincl
+*.xmpi
+
+# xindy
+*.xdy
+
+# xypic precompiled matrices
+*.xyc
+
+# endfloat
+*.ttt
+*.fff
+
+# Latexian
+TSWLatexianTemp*
+
+## Editors:
+# WinEdt
+*.bak
+*.sav
+
+# Texpad
+.texpadtmp
+
+# LyX
+*.lyx~
+
+# Kile
+*.backup
+
+# KBibTeX
+*~[0-9]*
+
+# auto folder when using emacs and auctex
+./auto/*
+*.el
+
+# expex forward references with \gathertags
+*-tags.tex
+
+# standalone packages
+*.sta

index.org

@@ -11,9 +11,15 @@
 
 #+HTML_HEAD: <link rel="stylesheet" type="text/css" href="https://research.tdehaeze.xyz/css/style.css"/>
 #+HTML_HEAD: <script type="text/javascript" src="https://research.tdehaeze.xyz/js/script.js"></script>
-
 #+HTML_MATHJAX: align: center tagside: right font: TeX
 
+#+BIND: org-latex-image-default-option "scale=1"
+#+BIND: org-latex-image-default-width ""
+
+#+LaTeX_CLASS: scrreprt
+#+LaTeX_CLASS_OPTIONS: [a4paper, 10pt, DIV=12]
+#+LaTeX_HEADER_EXTRA: \input{preamble.tex}
+
 #+CSL_STYLE: ieee.csl
 
 #+PROPERTY: header-args:matlab  :session *MATLAB*
@@ -41,7 +47,160 @@
 #+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png")
 :END:
 
-* Introduction                                                        :ignore:
+* LaTeX Config                                                      :noexport:
+#+begin_src emacs-lisp :tangle no
+  (setq org-latex-default-table-environment "tabular")
+  (setq org-latex-tables-booktabs nil)
+#+end_src
+
+#+begin_src latex :tangle preamble.tex
+  \usepackage{float}
+#+end_src
+
+#+begin_src latex :tangle preamble.tex
+  \usepackage{caption,tabularx,booktabs}
+#+end_src
+
+#+begin_src latex :tangle preamble.tex
+  \usepackage{biblatex}
+  \addbibresource{ref.bib}
+#+end_src
+
+#+begin_src latex :tangle preamble.tex
+  \usepackage{fontawesome}
+#+end_src
+
+#+begin_src latex :tangle preamble.tex
+  \usepackage{caption}
+  \usepackage{subcaption}
+
+  \captionsetup[figure]{labelfont=bf}
+  \captionsetup[subfigure]{labelfont=bf}
+  \captionsetup[listing]{labelfont=bf}
+  \captionsetup[table]{labelfont=bf}
+#+end_src
+
+#+begin_src latex :tangle preamble.tex
+  \usepackage{xcolor}
+
+  \definecolor{my-blue}{HTML}{6b7adb}
+  \definecolor{my-pale-blue}{HTML}{e6e9f9}
+  \definecolor{my-red}{HTML}{db6b6b}
+  \definecolor{my-pale-red}{HTML}{f9e6e6}
+  \definecolor{my-green}{HTML}{6bdbb6}
+  \definecolor{my-pale-green}{HTML}{e6f9f3}
+  \definecolor{my-yellow}{HTML}{dbd26b}
+  \definecolor{my-pale-yellow}{HTML}{f9f7e6}
+  \definecolor{my-orange}{HTML}{dba76b}
+  \definecolor{my-pale-orange}{HTML}{f9f0e6}
+  \definecolor{my-grey}{HTML}{a3a3a3}
+  \definecolor{my-pale-grey}{HTML}{f0f0f0}
+  \definecolor{my-turq}{HTML}{6bc7db}
+  \definecolor{my-pale-turq}{HTML}{e6f6f9}
+#+end_src
+
+#+begin_src latex :tangle preamble.tex
+  \usepackage{inconsolata}
+
+  \usepackage[newfloat=true, chapter]{minted}
+  \usemintedstyle{autumn}
+
+  \setminted{frame=lines,breaklines=true,tabsize=4,fontsize=\scriptsize,autogobble=true,labelposition=topline,bgcolor=my-pale-grey}
+  \setminted[matlab]{label=Matlab}
+  \setminted[latex]{label=LaTeX}
+  \setminted[bash]{label=Bash}
+  \setminted[python]{label=Python}
+  \setminted[text]{label=Results}
+
+  \setmintedinline{fontsize=\normalsize,bgcolor=my-pale-grey}
+#+end_src
+
+#+begin_src latex :tangle preamble.tex
+  \usepackage[most]{tcolorbox}
+
+  \tcbuselibrary{minted}
+
+  \newtcolorbox{seealso}[1][]{   enhanced,breakable,colback=my-pale-grey,colframe=my-grey,fonttitle=\bfseries,title=See Also,#1}
+  \newtcolorbox{hint}[1][]{      enhanced,breakable,colback=my-pale-grey,colframe=my-grey,fonttitle=\bfseries,title=Hint,#1}
+  \newtcolorbox{definition}[1][]{enhanced,breakable,colback=my-pale-red, colframe=my-red, fonttitle=\bfseries,title=Definition,#1}
+  \newtcolorbox{important}[1][]{ enhanced,breakable,colback=my-pale-red, colframe=my-red, fonttitle=\bfseries,title=Important,#1}
+  \newtcolorbox{exampl}[1][]{    enhanced,breakable,colback=my-pale-green,colframe=my-green,fonttitle=\bfseries,title=Example,#1}
+  \newtcolorbox{exercice}[1][]{  enhanced,breakable,colback=my-pale-yellow,colframe=my-yellow,fonttitle=\bfseries,title=Exercice,#1}
+  \newtcolorbox{question}[1][]{  enhanced,breakable,colback=my-pale-yellow,colframe=my-yellow,fonttitle=\bfseries,title=Question,#1}
+  \newtcolorbox{answer}[1][]{    enhanced,breakable,colback=my-pale-turq,colframe=my-turq,fonttitle=\bfseries,title=Answer,#1}
+  \newtcolorbox{summary}[1][]{   enhanced,breakable,colback=my-pale-blue,colframe=my-blue,fonttitle=\bfseries,title=Summary,#1}
+  \newtcolorbox{caution}[1][]{   enhanced,breakable,colback=my-pale-orange,colframe=my-orange,fonttitle=\bfseries,title=Caution,#1}
+  \newtcolorbox{warning}[1][]{   enhanced,breakable,colback=my-pale-orange,colframe=my-orange,fonttitle=\bfseries,title=Warning,#1}
+#+end_src
+
+#+begin_src latex :tangle preamble.tex
+  \newtcolorbox{my-quote}[1]{%
+    colback=my-pale-grey,
+    grow to right by=-10mm,
+    grow to left by=-10mm,
+    boxrule=0pt,
+    boxsep=0pt,
+    breakable,
+    enhanced jigsaw,
+    borderline west={4pt}{0pt}{my-grey}}
+
+  \renewenvironment{quote}{\begin{my-quote}}{\end{my-quote}}
+#+end_src
+
+#+begin_src latex :tangle preamble.tex
+  \newtcolorbox{my-verse}[1]{%
+    colback=my-pale-grey,
+    grow to right by=-10mm,
+    grow to left by=-10mm,
+    boxrule=0pt,
+    boxsep=0pt,
+    breakable,
+    enhanced jigsaw,
+    borderline west={4pt}{0pt}{my-grey}}
+
+  \renewenvironment{verse}{\begin{my-verse}}{\end{my-verse}}
+#+end_src
+
+#+begin_src latex :tangle preamble.tex
+  \usepackage{environ}% http://ctan.org/pkg/environ
+  \NewEnviron{aside}{%
+    \marginpar{\BODY}
+  }
+#+end_src
+
+#+begin_src latex :tangle preamble.tex
+  \renewenvironment{verbatim}{\VerbatimEnvironment\begin{minted}[]{text}}{\end{minted}}
+#+end_src
+
+#+begin_src latex :tangle preamble.tex
+  \usepackage{soul}
+  \sethlcolor{my-pale-grey}
+
+  \let\OldTexttt\texttt
+  \renewcommand{\texttt}[1]{{\ttfamily\hl{\mbox{\,#1\,}}}}
+#+end_src
+
+#+begin_src latex :tangle preamble.tex
+  \makeatletter
+  \preto\Gin@extensions{png,}
+  \DeclareGraphicsRule{.png}{pdf}{.pdf}{\noexpand\Gin@base.pdf}
+  \makeatother
+#+end_src
+
+#+begin_src latex :tangle preamble.tex
+  \usepackage{hyperref}
+  \hypersetup{
+    colorlinks = true,
+    allcolors = my-blue
+  }
+
+  \usepackage{hypcap}
+#+end_src
+
+* Introduction
+:PROPERTIES:
+:UNNUMBERED: t
+:END:
 
 The purpose of this document is to give a /practical introduction/ to the wonderful world of $\mathcal{H}_\infty$ Control.
 
@@ -78,12 +237,6 @@ The general structure of this document is as follows:
 * Introduction to Model Based Control
 <<sec:model_based_control>>
 
-** Introduction                                                      :ignore:
-
-# - Section [[sec:model_based_control_methodology]]
-# - Section [[sec:comp_classical_modern_robust_control]]
-# - Section [[sec:example_system]]
-
 ** Model Based Control - Methodology
 <<sec:model_based_control_methodology>>
 
@@ -160,32 +313,32 @@ The three presented control methods are compared in Table [[tab:comparison_contr
 
 Note that in parallel, there have been numerous other developments, including non-linear control, adaptive control, machine-learning control just to name a few.
 
+
 #+name: tab:comparison_control_methods
 #+caption: Table summurazing the main differences between classical, modern and robust control
-| <l>                     |                <c>                 |                 <c>                  |                    <c>                     |
+#+attr_latex: :environment tabularx :booktabs t :width \linewidth :align lccc
-|                         |        *Classical Control*         |           *Modern Control*           |              *Robust Control*              |
+| <l>           |         <c>         |           <c>            |              <c>              |
-|-------------------------+------------------------------------+--------------------------------------+--------------------------------------------|
+|               | *Classical Control* |     *Modern Control*     |       *Robust Control*        |
-| *Date*                  |               1930-                |                1960-                 |                   1980-                    |
+|---------------+---------------------+--------------------------+-------------------------------|
-|-------------------------+------------------------------------+--------------------------------------+--------------------------------------------|
+| *Date*        |        1930-        |          1960-           |             1980-             |
-| *Tools*                 |         Transfer Functions         |       State Space formulation        |         Systems and Signals Norms          |
+|---------------+---------------------+--------------------------+-------------------------------|
-|                         |        Nyquist, Bode Plots         |          Riccati Equations           |       Closed Loop Transfer Functions       |
+| *Tools*       | Transfer Functions  | State Space formulation  |     Systems/Signal Norms      |
-|                         |             Root Locus             |                                      |            Closed Loop Shaping             |
+|               | Nyquist, Bode Plots |    Riccati Equations     |        Closed Loop TF         |
-|                         |       Phase and Gain margins       |                                      |            Weighting Functions             |
+|               |     Root Locus      |      Kalman Filters      |      Closed Loop Shaping      |
-|                         |         Open Loop Shaping          |                                      |                Disk margin                 |
+|               | Phase/Gain margins  |                          |      Weighting Functions      |
-|                         |                                    |                                      |        Singular Value Decomposition        |
+|               |  Open Loop Shaping  |                          |          Disk margin          |
-|-------------------------+------------------------------------+--------------------------------------+--------------------------------------------|
+|---------------+---------------------+--------------------------+-------------------------------|
-| *Control Architectures* | Proportional, Integral, Derivative |       Full State Feedback, LQR       |       General Control Configuration        |
+| *Controllers* |     P, PI, PID      |   Full State Feedback    |     General Control Conf.     |
-|                         |            Leads, Lags             |         Kalman Filters, LQG          |             Generalized Plant              |
+|               |     Leads, Lags     |         LQG, LQR         |                               |
-|-------------------------+------------------------------------+--------------------------------------+--------------------------------------------|
+|---------------+---------------------+--------------------------+-------------------------------|
-| *Advantages*            |          Study Stability           |         Automatic Synthesis          |            Automatic Synthesis             |
+| *Advantages*  |   Study Stability   |   Automatic Synthesis    |      Automatic Synthesis      |
-|                         |               Simple               |                 MIMO                 |                    MIMO                    |
+|               |       Simple        |           MIMO           |             MIMO              |
-|                         |              Natural               |         Optimization Problem         |            Optimization Problem            |
+|               |       Natural       |   Optimization Problem   |     Optimization Problem      |
-|                         |                                    |                                      |           Guaranteed Robustness            |
+|               |                     |                          |     Guaranteed Robustness     |
-|                         |                                    |                                      |     Easy specification of performances     |
+|---------------+---------------------+--------------------------+-------------------------------|
-|-------------------------+------------------------------------+--------------------------------------+--------------------------------------------|
+| *Disadvant.*  |    Manual Method    | No Guaranteed Robustness |  Requires knowledge of tools  |
-| *Disadvantages*         |           Manual Method            |       No Guaranteed Robustness       |    Required knowledge of specific tools    |
+|               |      Only SISO      |    Rejection of Pert.    | Need good model of the system |
-|                         |             Only SISO              | Difficult Rejection of Perturbations | Need a reasonably good model of the system |
+|               | No input usage lim. |                          |                               |
-|                         | No clear way to limit input usage  |                                      |                                            |
 
 #+begin_src latex :file robustness_performance.pdf
   \begin{tikzpicture}
@@ -315,6 +468,7 @@ The notations used on Figure [[fig:mech_sys_1dof_inertial_contr]] are listed and
 
 #+name: tab:example_notations
 #+caption: Example system variables
+#+attr_latex: :environment tabularx :booktabs t :width \linewidth :align cXcc
 | *Notation*         | *Description*                                                  | *Value*        | *Unit*    |
 |--------------------+----------------------------------------------------------------+----------------+-----------|
 | $m$                | Payload's mass to position / isolate                           | $10$           | [kg]      |
@@ -335,6 +489,7 @@ Derive the following open-loop transfer functions:
   \end{align}
 
 #+HTML: <details><summary>Hint</summary>
+#+LATEX: \tcbsubtitle{Hint}
 You can follow this generic procedure:
 1. List all applied forces ot the mass: Actuator force, Stiffness force (Hooke's law), ...
 2. Apply the Newton's Second Law on the payload
@@ -345,6 +500,7 @@ You can follow this generic procedure:
 #+HTML: </details>
 
 #+HTML: <details><summary>Results</summary>
+#+LATEX: \tcbsubtitle{Results}
 \begin{align}
   G(s)   &= \frac{1}{m s^2 + cs + k} \\
   G_d(s) &= \frac{cs + k}{m s^2 + cs + k}
@@ -501,6 +657,7 @@ Its name comes from the fact that this is actually the "gain around the loop".
 
 #+name: fig:open_loop_shaping
 #+caption: Classical Feedback Architecture
+#+attr_latex: :float nil
 [[file:figs/open_loop_shaping.png]]
 #+end_definition
 
@@ -584,6 +741,7 @@ Using =SISOTOOL=, design a controller that fulfills the specifications.
 #+end_src
 
 #+HTML: <details><summary>Hint</summary>
+#+LATEX: \tcbsubtitle{Hint}
 You can follow this procedure:
 1. In order to have good disturbance rejection at low frequency, add a simple or double *integrator*
 2. In terms of the loop gain, the *bandwidth* can be defined at the frequency $\omega_c$ where $|l(j\omega_c)|$ first crosses 1 from above.
@@ -687,12 +845,13 @@ Such shape corresponds to the typical wanted Loop gain Shape shown in Figure [[f
 
 #+name: tab:open_loop_shaping_specifications
 #+caption: Wanted Loop Shape corresponding to each specification
-|                         | Specification                               | Corresponding Loop Shape                                        |
+#+attr_latex: :environment tabularx :booktabs t :width \linewidth :align lXX
-|-------------------------+---------------------------------------------+-----------------------------------------------------------------|
+|              | *Specification*                        | *Corresponding Loop Shape*               |
-| *Disturbance Rejection* | Highest possible rejection below 1Hz        | Slope of -40dB/decade at low frequency to have a high loop gain |
+|--------------+----------------------------------------+------------------------------------------|
-| *Positioning Speed*     | Bandwidth of approximately 10Hz             | $L$ crosses 1 at 10Hz: $\vert L_w(j2 \pi 10)\vert = 1$          |
+| *Dist. Rej.* | Highest possible rejection below 1Hz   | Slope of -40dB/dec at low frequency      |
-| *Noise Attenuation*     | Roll-off of -40dB/decade past 30Hz          | Roll-off of -40dB/decade past 30Hz                              |
+| *Pos. Speed* | Bandwidth of approximately 10Hz        | $L$ crosses 1 at 10Hz                    |
-| *Robustness*            | Gain margin > 3dB and Phase margin > 30 deg | Slope of -20dB/decade near the crossover                        |
+| *Noise Att.* | Roll-off of -40dB/decade past 30Hz     | Roll-off of -40dB/decade past 30Hz       |
+| *Robustness* | $\Delta G > 3dB$, $\Delta \phi > 30^o$ | Slope of -20dB/decade near the crossover |
 
 Then, a (stable, minimum phase) transfer function $L_w(s)$ should be created that has the same gain as the wanted shape of the Loop gain.
 For this example, a double integrator and a lead centered on 10Hz are used.
@@ -808,12 +967,13 @@ Let's finally compare the obtained stability margins of the $\mathcal{H}_\infty$
 
 #+name: tab:open_loop_shaping_compare
 #+caption: Comparison of the characteristics obtained with the two methods
+#+attr_latex: :environment tabularx :booktabs t :width \linewidth :align Xcc
 #+RESULTS:
-| Specifications              | Manual Method | $\mathcal{H}_\infty$ Method |
+| *Specifications*            | *Manual Method* | *$\mathcal{H}_\infty$ Method* |
-|-----------------------------+---------------+-----------------------------|
+|-----------------------------+-----------------+-------------------------------|
-| Gain Margin $> 3$ [dB]      |           3.1 |                        31.7 |
+| Gain Margin $> 3$ [dB]      |             3.1 |                          31.7 |
-| Phase Margin $> 30$ [deg]   |          35.4 |                        54.7 |
+| Phase Margin $> 30$ [deg]   |            35.4 |                          54.7 |
-| Crossover $\approx 10$ [Hz] |          10.1 |                         9.9 |
+| Crossover $\approx 10$ [Hz] |            10.1 |                           9.9 |
 
 * A first Step into the $\mathcal{H}_\infty$ world
 <<sec:h_infinity_introduction>>
@@ -873,6 +1033,7 @@ We can see in Figure [[fig:hinfinity_norm_siso_bode]] that indeed, the $\mathcal
 
 #+name: fig:hinfinity_norm_siso_bode
 #+caption: Example of the $\mathcal{H}_\infty$ norm of a SISO system
+#+attr_latex: :float nil
 #+RESULTS:
 [[file:figs/hinfinity_norm_siso_bode.png]]
 #+end_exampl
@@ -941,18 +1102,20 @@ A practical example about how to derive the generalized plant for a classical co
 #+begin_important
 #+name: fig:general_plant
 #+caption: Inputs and Outputs of the generalized Plant
+#+ATTR_LATEX: :float nil
 #+RESULTS:
 [[file:figs/general_plant.png]]
 
 #+name: tab:notation_general
 #+caption: Notations for the general configuration
-| Notation | Meaning                                            |
+#+attr_latex: :environment tabularx :booktabs t :width 0.8\linewidth :align cX :float nil
-|----------+----------------------------------------------------|
+| *Notation* | *Meaning*                                          |
-| $P$      | Generalized plant model                            |
+|------------+----------------------------------------------------|
-| $w$      | Exogenous inputs: references, disturbances, noises |
+| $P$        | Generalized plant model                            |
-| $z$      | Exogenous outputs: signals to be minimized         |
+| $w$        | Exogenous inputs: references, disturbances, noises |
-| $v$      | Controller inputs: measurements                    |
+| $z$        | Exogenous outputs: signals to be minimized         |
-| $u$      | Control signals                                    |
+| $v$        | Controller inputs: measurements                    |
+| $u$        | Control signals                                    |
 #+end_important
 
 ** The $\mathcal{H}_\infty$ Synthesis applied on the Generalized plant
@@ -992,6 +1155,7 @@ Once the generalized plant is obtained, the $\mathcal{H}_\infty$ synthesis probl
 
 #+name: fig:general_control_names
 #+caption: General Control Configuration
+#+attr_latex: :float nil
 #+RESULTS:
 [[file:figs/general_control_names.png]]
 #+end_important
@@ -1067,24 +1231,28 @@ The procedure to convert a typical control architecture as the one shown in Figu
 
   #+name: fig:classical_feedback_tracking
   #+caption: Classical Feedback Control Architecture (Tracking)
+  #+attr_latex: :float nil
   [[file:figs/classical_feedback_tracking.png]]
 
 #+HTML: <details><summary>Hint</summary>
+#+LATEX: \tcbsubtitle{Hint}
   First, define the signals of the generalized plant:
   - Exogenous inputs: $w = r$
   - Signals to be minimized:
     Usually, we want to minimize the tracking errors $\epsilon$ and the control signal $u$: $z = [\epsilon,\ u]$
   - Controller inputs: this is the signal at the input of the controller: $v = \epsilon$
-  - Control inputs: signal generated by the controller: $u$
+  - Controller outputs: signal generated by the controller: $u$
 
   Then, Remove $K$ and rearrange the inputs and outputs as in Figure [[fig:general_plant]].
 #+HTML: </details>
 
 #+HTML: <details><summary>Answer</summary>
+#+LATEX: \tcbsubtitle{Anwser}
   The obtained generalized plant shown in Figure [[fig:mixed_sensitivity_ref_tracking]].
 
   #+name: fig:mixed_sensitivity_ref_tracking
   #+caption: Generalized plant of the Classical Feedback Control Architecture (Tracking)
+  #+attr_latex: :float nil
   [[file:figs/mixed_sensitivity_ref_tracking.png]]
 
   Using Matlab, the generalized plant can be defined as follows:
@@ -1161,7 +1329,8 @@ These are summarized in Table [[tab:spec_closed_loop_tf]].
 
 #+name: tab:spec_closed_loop_tf
 #+caption: Typical Specification and associated closed-loop transfer function
-| Specification                  | Closed-Loop Transfer Function                 |
+#+attr_latex: :environment tabularx :booktabs t :width 0.8\linewidth :align Xl
+| *Specification*                | *CL Transfer Function*                        |
 |--------------------------------+-----------------------------------------------|
 | Reference Tracking             | From $r$ to $\epsilon$                        |
 | Disturbance Rejection          | From $d$ to $y$                               |
@@ -1174,13 +1343,14 @@ These are summarized in Table [[tab:spec_closed_loop_tf]].
 For the feedback system in Figure [[fig:gang_of_four_feedback]], write the output signals $[\epsilon, u, y]$ as a function of the systems $K(s), G(s)$ and the input signals $[r, d, n]$.
 
 #+HTML: <details><summary>Hint</summary>
+#+LATEX: \tcbsubtitle{Hint}
 Take one of the output (e.g. $y$), and write it as a function of the inputs $[d, r, n]$ going step by step around the loop:
-\begin{aligned}
+\begin{align*}
   y &= G u \\
     &= G (d + K \epsilon) \\
     &= G \big(d + K (r - n - y) \big) \\
     &= G d + GK r - GK n - GK y
-\end{aligned}
+\end{align*}
 
 Isolate $y$ at the right hand side, and finally obtain:
 \[ y = \frac{GK}{1+ GK} r + \frac{G}{1 + GK} d - \frac{GK}{1 + GK} n \]
@@ -1189,6 +1359,7 @@ Do the same procedure for $u$ and $\epsilon$
 #+HTML: </details>
 
 #+HTML: <details><summary>Answer</summary>
+#+LATEX: \tcbsubtitle{Answer}
 The following equations should be obtained:
 \begin{align}
   y        &= \frac{GK}{1 + GK} r + \frac{G}{1 + GK} d - \frac{GK}{1 + GK} n \\
@@ -1238,11 +1409,12 @@ The comparison of the sensitivity functions shapes and their effect on the step
 
 #+name: tab:compare_sensitivity_shapes
 #+caption: Comparison of the sensitivity function shape and the corresponding step response for the three controller variations
-| Controller | Sensitivity Function Shape                         | Change of the Step Response      |
+#+attr_latex: :environment tabularx :booktabs t :width 0.9\linewidth :align lXX
-|------------+----------------------------------------------------+----------------------------------|
+| *Controller* | *Sensitivity Function Shape*                       | *Change of the Step Response*    |
-| $K_1(s)$   | Larger bandwidth $\omega_b$                        | Faster rise time                 |
+|--------------+----------------------------------------------------+----------------------------------|
-| $K_2(s)$   | Larger peak value $\Vert S\Vert_\infty$            | Large overshoot and oscillations |
+| $K_1(s)$     | Larger bandwidth $\omega_b$                        | Faster rise time                 |
-| $K_3(s)$   | Larger low frequency gain $\vert S(j\cdot 0)\vert$ | Larger static error              |
+| $K_2(s)$     | Larger peak value $\Vert S\Vert_\infty$            | Large overshoot and oscillations |
+| $K_3(s)$     | Larger low frequency gain $\vert S(j\cdot 0)\vert$ | Larger static error              |
 
 #+begin_src matlab :exports none
   wc = 2*pi*1; L_w = 8; wi = 2*pi*0.02;
@@ -1379,6 +1551,7 @@ From the simple analysis above, we can draw a first estimation of the wanted sha
 
 #+name: fig:h-infinity-spec-S
 #+caption: Typical wanted shape of the Sensitivity transfer function
+#+attr_latex: :float nil
 [[file:figs/h-infinity-spec-S.png]]
 #+end_important
 
@@ -1445,10 +1618,10 @@ Or does it? Let's find out.
 
 #+name: fig:phase_gain_margin_model_plant
 #+caption: Bode plot of the obtained Loop Gain $L(s)$
+#+attr_latex: :float nil
 #+RESULTS:
 [[file:figs/phase_gain_margin_model_plant.png]]
 
-
 Now let's suppose the controller is implemented in practice, and the "real" plant $G_r(s)$ as a slightly lower damping factor than the one estimated for the model:
 #+begin_src matlab
   xi = 0.03;
@@ -1504,6 +1677,7 @@ It is confirmed by checking the stability of the closed loop system:
 
 #+name: fig:phase_gain_margin_real_plant
 #+caption: Bode plots of $L(s)$ (loop gain corresponding the nominal plant) and $L_r(s)$ (loop gain corresponding to the real plant)
+#+attr_latex: :float nil
 #+RESULTS:
 [[file:figs/phase_gain_margin_real_plant.png]]
 
@@ -1516,7 +1690,7 @@ Let's now determine a new robustness indicator based on the Nyquist Stability Cr
 
 #+begin_definition
 - Nyquist Stability Criteria (for stable systems) ::
-  If the open-loop transfer function $L(s)$ is stable, then the closed-loop system will be unstable for any encirclement of the point $−1$ on the Nyquist plot.
+  If the open-loop transfer function $L(s)$ is stable, then the closed-loop system will be unstable for any encirclement of the point $-1$ on the Nyquist plot.
 
 - Nyquist Plot ::
   The Nyquist plot shows the evolution of $L(j\omega)$ in the complex plane from $\omega = 0 \to \infty$.
@@ -1591,6 +1765,7 @@ The gain, phase and module margins are graphically shown to have an idea of what
 
 #+name: fig:module_margin_example
 #+caption: Nyquist plot with visual indication of the Gain margin $\Delta G$, Phase margin $\Delta \phi$ and Module margin $\Delta M$
+#+attr_latex: :float nil
 #+RESULTS:
 [[file:figs/module_margin_example.png]]
 #+end_exampl
@@ -1642,13 +1817,14 @@ And we now understand why setting an upper bound on the magnitude of $S$ is gene
 
 #+name: tab:specification_modern
 #+caption: Typical Specifications and corresponding wanted norms of open and closed loop tansfer functions
-|                             | Open-Loop Shaping  | Closed-Loop Shaping                        |
+#+attr_latex: :environment tabularx :booktabs t :width 0.9\linewidth :align lXX
-|-----------------------------+--------------------+--------------------------------------------|
+|                             | *Open-Loop Shaping* | *Closed-Loop Shaping*                      |
-| Reference Tracking          | $L$ large          | $S$ small                                  |
+|-----------------------------+---------------------+--------------------------------------------|
-| Disturbance Rejection       | $L$ large          | $GS$ small                                 |
+| Reference Tracking          | $L$ large           | $S$ small                                  |
-| Measurement Noise Filtering | $L$ small          | $T$ small                                  |
+| Disturbance Rejection       | $L$ large           | $GS$ small                                 |
-| Small Command Amplitude     | $K$ and $L$ small  | $KS$ small                                 |
+| Measurement Noise Filtering | $L$ small           | $T$ small                                  |
-| Robustness                  | Phase/Gain margins | Module margin: $\Vert S\Vert_\infty$ small |
+| Small Command Amplitude     | $K$ and $L$ small   | $KS$ small                                 |
+| Robustness                  | Phase/Gain margins  | Module margin: $\Vert S\Vert_\infty$ small |
 
 * $\mathcal{H}_\infty$ Shaping of closed-loop transfer functions
 <<sec:closed-loop-shaping>>
@@ -1769,12 +1945,14 @@ Let's now show how this is equivalent as *shaping* the sensitivity function:
 Using matlab, compute the weighted generalized plant shown in Figure [[fig:first_order_weight]] as a function of $G(s)$ and $W_S(s)$.
 
 #+HTML: <details><summary>Hint</summary>
+#+LATEX: \tcbsubtitle{Hint}
 The weighted generalized plant can be defined in Matlab using two techniques:
 - by writing manually the 4 transfer functions from $[w, u]$ to $[\tilde{\epsilon}, v]$
 - by pre-multiplying the (non-weighted) generalized plant by a block-diagonal transfer function matrix containing the weights for the outputs $z$ and =1= for the outputs $v$
 #+HTML: </details>
 
 #+HTML: <details><summary>Answer</summary>
+#+LATEX: \tcbsubtitle{Answer}
 The two solutions below can be used.
 
 #+begin_src matlab :tangle no :eval no
@@ -1823,7 +2001,7 @@ The Matlab code below produces a weighting function with the following character
 - Gain of 1 at 10Hz
 - High frequency gain of 0.5
 
-#+begin_src matlab
+#+begin_src matlab :float nil
   Ws = makeweight(1e2, [2*pi*10, 1], 1/2);
 #+end_src
 
@@ -1845,6 +2023,7 @@ The Matlab code below produces a weighting function with the following character
 
 #+name: fig:first_order_weight
 #+caption: Obtained Magnitude of the Weighting Function
+#+attr_latex: :float nil
 #+RESULTS:
 [[file:figs/first_order_weight.png]]
 #+end_exampl
@@ -1870,8 +2049,6 @@ The parameters permit to specify:
 
 A Matlab function implementing Equation eqref:eq:weight_formula_advanced is shown below:
 
-#+name: lst:generateWeight
-#+caption: Matlab Function that can be used to generate Weighting functions
 #+begin_src matlab :tangle matlab/generateWeight.m :comments none :eval no
   function [W] = generateWeight(args)
       arguments
@@ -1890,7 +2067,7 @@ A Matlab function implementing Equation eqref:eq:weight_formula_advanced is show
 
       s = zpk('s');
 
-      W = (((1/args.wc)*sqrt((1-(args.G0/args.Gc)^(2/args.n))/(1-(args.Gc/args.G1)^(2/args.n)))*s + (args.G0/args.Gc)^(1/args.n))/((1/args.G1)^(1/args.n)*(1/args.wc)*sqrt((1-(args.G0/args.Gc)^(2/args.n))/(1-(args.Gc/args.G1)^(2/args.n)))*s + (1/args.Gc)^(1/args.n)))^args.n;
+      W = (((1/args.wc) * sqrt((1-(args.G0/args.Gc)^(2/args.n))/(1-(args.Gc/args.G1)^(2/args.n)))*s + (args.G0/args.Gc)^(1/args.n)) / ((1/args.G1)^(1/args.n) * (1/args.wc) * sqrt((1-(args.G0/args.Gc)^(2/args.n))/(1-(args.Gc/args.G1)^(2/args.n)))*s + (1/args.Gc)^(1/args.n)))^args.n;
 
   end
 #+end_src
@@ -1928,6 +2105,7 @@ The obtained shapes are shown in Figure [[fig:high_order_weight]].
 
 #+name: fig:high_order_weight
 #+caption: Higher order weights using Equation eqref:eq:weight_formula_advanced
+#+attr_latex: :float nil
 #+RESULTS:
 [[file:figs/high_order_weight.png]]
 #+end_seealso
@@ -1946,6 +2124,7 @@ As usual, the plant used is the one presented in Section [[sec:example_system]].
 Translate the requirements as upper bounds on the Sensitivity function and design the corresponding weighting functions using Matlab.
 
 #+HTML: <details><summary>Hint</summary>
+#+LATEX: \tcbsubtitle{Hint}
 The typical wanted upper bound of the sensitivity function is shown in Figure [[fig:h-infinity-spec-S-bis]].
 
 More precisely:
@@ -1958,10 +2137,12 @@ Remember that the wanted upper bound of the sensitivity function is defined by t
 
 #+name: fig:h-infinity-spec-S-bis
 #+caption: Typical wanted shape of the Sensitivity transfer function
+#+attr_latex: :float nil
 [[file:figs/h-infinity-spec-S.png]]
 #+HTML: </details>
 
 #+HTML: <details><summary>Answer</summary>
+#+LATEX: \tcbsubtitle{Answer}
 We want to design the weighting function $W_s(s)$ such that:
 1. $|W_s(j \cdot 2 \pi 10)| = \sqrt{2}$
 2. $|W_s(j \cdot 0)| = 10^3$
@@ -2016,10 +2197,10 @@ And the $\mathcal{H}_\infty$ synthesis is performed on the /weighted/ generalize
 #+end_example
 
 $\gamma \approx 0.5$ means that the $\mathcal{H}_\infty$ synthesis generated a controller $K(s)$ that stabilizes the closed-loop system, and such that the $\mathcal{H}_\infty$ norm of the closed-loop transfer function from $w$ to $z$ is less than $\gamma$:
-\begin{aligned}
+\begin{align*}
   & \| W_s(s) S(s) \|_\infty \approx 0.5 \\
   & \Leftrightarrow |S(j\omega)| < \frac{0.5}{|W_s(j\omega)|} \quad \forall \omega
-\end{aligned}
+\end{align*}
 
 This is indeed what we can see by comparing $|S|$ and $|W_S|$ in Figure [[fig:results_sensitivity_hinf]].
 
@@ -2063,31 +2244,33 @@ When multiple closed-loop transfer function are shaped at the same time, it is r
 
 #+name: tab:usual_shaping_gang_four
 #+caption: Typical specifications and corresponding shaping of the /Gang of four/
-| <l>                                  |     <c>      | <l>                                                       |
+#+attr_latex: :environment tabularx :booktabs t :width 0.9\linewidth :align llX
-| Specification                        | Gang of Four | Wanted shape                                              |
+| <l>                           | <c>  | <l>                                            |
-|--------------------------------------+--------------+-----------------------------------------------------------|
+| *Specifications*              | *TF* | *Wanted shape*                                 |
-| Fast Reference Tracking              |     $S$      | Set lower bound on the bandwidth                          |
+|-------------------------------+------+------------------------------------------------|
-| Small Steady State Errors            |     $S$      | Small low frequency gain                                  |
+| Fast Reference Tracking       | $S$  | Set lower bound on the bandwidth               |
-| Follow Step ref. inputs              |     $S$      | Slope of +20dB/dec at low frequency                       |
+| Small Steady State Errors     | $S$  | Small low frequency gain                       |
-| Follow Ramp ref. inputs              |     $S$      | Slope of +40dB/dec at low frequency                       |
+| Follow Step ref. inputs       | $S$  | Slope of +20dB/dec at low frequency            |
-| Follow Sinusoidal ref. inputs        |     $S$      | Small magnitude centered on the sin. frequency            |
+| Follow Ramp ref. inputs       | $S$  | Slope of +40dB/dec at low frequency            |
-|--------------------------------------+--------------+-----------------------------------------------------------|
+| Follow Sin. ref. inputs       | $S$  | Small magnitude centered on the sin. frequency |
-| Output Disturbance Rejection         |     $S$      | Small gain in the disturbance bandwidth                   |
+|-------------------------------+------+------------------------------------------------|
-| Input Disturbance Rejection          |     $GS$     | Small gain in the disturbance bandwidth                   |
+| Output Disturbance Rejection  | $S$  | Small gain in the disturbance bandwidth        |
-| Prevent inversion of resonant plants |     $GS$     | Limit gain around resonance                               |
+| Input Disturbance Rejection   | $GS$ | Small gain in the disturbance bandwidth        |
-|--------------------------------------+--------------+-----------------------------------------------------------|
+| Prevent notching resonances   | $GS$ | Limit gain around resonance                    |
-| Small Command Amplitude              |     $KS$     | Small at high frequency                                   |
+|-------------------------------+------+------------------------------------------------|
-| Limitation of the Control Bandwidth  |     $T$      | Set an upper bound on the bandwidth                       |
+| Small Command Amplitude       | $KS$ | Small at high frequency                        |
-| Measurement Noise Filtering          |     $T$      | Small high frequency gain                                 |
+| Limitation of the Bandwidth   | $T$  | Set an upper bound on the bandwidth            |
-|--------------------------------------+--------------+-----------------------------------------------------------|
+| Measurement Noise Filtering   | $T$  | Small high frequency gain                      |
-| Stability margins                    |     $S$      | Module margin: $\Vert S\Vert_\infty$ small                |
+|-------------------------------+------+------------------------------------------------|
-| Robustness to un-modelled dynamics   |     $T$      | Small at frequencies where the plant uncertainty is large |
+| Stability margins             | $S$  | Module margin: $\Vert S\Vert_\infty$ small     |
+| Robust to unmodelled dynamics | $T$  | Small at freq. where uncertainty is large      |
 
 Depending on which closed-loop transfer function are to be shaped, different weighted generalized plant can be used.
 Some of them are described below for reference, it is a good exercise to try to re-design such weighted generalized plants.
 
 *** S KS                                                            :ignore:
 #+HTML: <details><summary>Shaping of S and KS</summary>
+#+latex: \begin{exampl}[after title={~- Shape $S$ and $KS$}]
 #+begin_src latex :file general_conf_shaping_S_KS.pdf
   \begin{tikzpicture}
     % Blocs
@@ -2126,6 +2309,7 @@ Some of them are described below for reference, it is a good exercise to try to
 
 #+name: fig:general_conf_shaping_S_KS
 #+caption: Generalized Plant to shape $S$ and $KS$
+#+attr_latex: :float nil
 #+RESULTS:
 [[file:figs/general_conf_shaping_S_KS.png]]
 
@@ -2133,17 +2317,18 @@ Weighting functions:
 - $W_1(s)$ is used to shape $S$
 - $W_2(s)$ is used to shape $KS$
 
-#+name: lst:general_plant_S_KS
-#+caption: General Plant definition corresponding to Figure [[fig:general_conf_shaping_S_KS]]
 #+begin_src matlab :eval no :tangle no
-  P = [W1 -G*W1
+  P = [1 -G
-       0   W2
+       0  1
-       1  -G];
+       1 -G];
+  Pw = blkdiag(W1, W2, 1)*P;
 #+end_src
+#+latex: \end{exampl}
 #+HTML: </details>
 
 *** S T                                                             :ignore:
 #+HTML: <details><summary>Shaping of S and T</summary>
+#+latex: \begin{exampl}[after title={~- Shape $S$ and $T$}]
 #+begin_src latex :file general_conf_shaping_S_T.pdf
   \begin{tikzpicture}
     % Blocs
@@ -2180,6 +2365,7 @@ Weighting functions:
 
 #+name: fig:general_conf_shaping_S_T
 #+caption: Generalized Plant to shape $S$ and $T$
+#+attr_latex: :float nil
 #+RESULTS:
 [[file:figs/general_conf_shaping_S_T.png]]
 
@@ -2187,17 +2373,18 @@ Weighting functions:
 - $W_1$ is used to shape $S$
 - $W_2$ is used to shape $T$
 
-#+name: lst:general_plant_S_T
-#+caption: General Plant definition corresponding to Figure [[fig:general_conf_shaping_S_T]]
 #+begin_src matlab :eval no :tangle no
-  P = [W1 -G*W1
+  P = [1 -G
-       0   G*W2
+       0  G
-       1   -G];
+       1 -G];
+  Pw = blkdiag(W1, W2, 1)*P;
 #+end_src
+#+latex: \end{exampl}
 #+HTML: </details>
 
 *** S GS                                                            :ignore:
 #+HTML: <details><summary>Shaping of S and GS</summary>
+#+latex: \begin{exampl}[after title={~- Shape $S$ and $GS$}]
 #+begin_src latex :file general_conf_shaping_S_GS.pdf
   \begin{tikzpicture}
     % Blocs
@@ -2234,6 +2421,7 @@ Weighting functions:
 
 #+name: fig:general_conf_shaping_S_GS
 #+caption: Generalized Plant to shape $S$ and $GS$
+#+attr_latex: :float nil
 #+RESULTS:
 [[file:figs/general_conf_shaping_S_GS.png]]
 
@@ -2241,17 +2429,18 @@ Weighting functions:
 - $W_1$ is used to shape $S$
 - $W_2$ is used to shape $GS$
 
-#+name: lst:general_plant_S_GS
-#+caption: General Plant definition corresponding to Figure [[fig:general_conf_shaping_S_GS]]
 #+begin_src matlab :eval no :tangle no
-  P = [W1   -W1
+  P = [1 -1
-       G*W2 -G*W2
+       G -G
-       G    -G];
+       G -G];
+  Pw = blkdiag(W1, W2, 1)*P;
 #+end_src
+#+latex: \end{exampl}
 #+HTML: </details>
 
 *** S T KS                                                          :ignore:
 #+HTML: <details><summary>Shaping of S, T and KS</summary>
+#+latex: \begin{exampl}[after title={~- Shape $S$, $T$ and $KS$}]
 #+begin_src latex :file general_conf_shaping_S_T_KS.pdf
   \begin{tikzpicture}
     % Blocs
@@ -2293,6 +2482,7 @@ Weighting functions:
 
 #+name: fig:general_conf_shaping_S_T_KS
 #+caption: Generalized Plant to shape $S$, $T$ and $KS$
+#+attr_latex: :float nil
 #+RESULTS:
 [[file:figs/general_conf_shaping_S_T_KS.png]]
 
@@ -2301,18 +2491,19 @@ Weighting functions:
 - $W_2$ is used to shape $KS$
 - $W_3$ is used to shape $T$
 
-#+name: lst:general_plant_S_T_KS
-#+caption: General Plant definition corresponding to Figure [[fig:general_conf_shaping_S_T_KS]]
 #+begin_src matlab :eval no :tangle no
-  P = [W1 -G*W1
+  P = [1 -G
-       0   W2
+       0  1
-       0   G*W3
+       0  G
-       1   -G];
+       1 -G];
+  Pw = blkdiag(W1, W2, W3, 1)*P;
 #+end_src
+#+latex: \end{exampl}
 #+HTML: </details>
 
 *** S T GS                                                          :ignore:
 #+HTML: <details><summary>Shaping of S, T and GS</summary>
+#+latex: \begin{exampl}[after title={~- Shape $S$, $T$ and $GS$}]
 #+begin_src latex :file general_conf_shaping_S_T_GS.pdf
   \begin{tikzpicture}
     % Blocs
@@ -2354,6 +2545,7 @@ Weighting functions:
 
 #+name: fig:general_conf_shaping_S_T_GS
 #+caption: Generalized Plant to shape $S$, $T$ and $GS$
+#+attr_latex: :float nil
 #+RESULTS:
 [[file:figs/general_conf_shaping_S_T_GS.png]]
 
@@ -2362,18 +2554,19 @@ Weighting functions:
 - $W_2$ is used to shape $GS$
 - $W_3$ is used to shape $T$
 
-#+name: lst:general_plant_S_T_GS
-#+caption: General Plant definition corresponding to Figure [[fig:general_conf_shaping_S_T_GS]]
 #+begin_src matlab :eval no :tangle no
-  P = [W1   -W1
+  P = [1 -1
-       G*W2 -G*W2
+       G -G
-       0     W3
+       0  1
-       G    -G];
+       G -G];
+  Pw = blkdiag(W1, W2, W3, 1)*P;
 #+end_src
+#+latex: \end{exampl}
 #+HTML: </details>
 
 *** S T KS GS                                                       :ignore:
 #+HTML: <details><summary>Shaping of S, T, KS and GS</summary>
+#+latex: \begin{exampl}[after title={~- Shape $S$, $T$, $KS$ and $GS$}]
 #+begin_src latex :file general_conf_shaping_S_T_KS_GS.pdf
   \begin{tikzpicture}
     % Blocs
@@ -2419,6 +2612,7 @@ Weighting functions:
 
 #+name: fig:general_conf_shaping_S_T_KS_GS
 #+caption: Generalized Plant to shape $S$, $T$, $KS$ and $GS$
+#+attr_latex: :float nil
 #+RESULTS:
 [[file:figs/general_conf_shaping_S_T_KS_GS.png]]
 
@@ -2428,13 +2622,13 @@ Weighting functions:
 - $W_1W_3$ is used to shape $GS$
 - $W_2W_3$ is used to shape $T$
 
-#+name: lst:general_plant_S_T_KS_GS
-#+caption: General Plant definition corresponding to Figure [[fig:general_conf_shaping_S_T_KS_GS]]
 #+begin_src matlab :eval no :tangle no
-  P = [ W1  -W1*G*W3 -G*W1
+  P = [ 1 -G -G
-        0    0        W2
+        0  0  1
-        1   -G*W3    -G];
+        1 -G -G];
+  Pw = blkdiag(W1, W2, 1)*P*blkdiag(1, W3, 1);
 #+end_src
+#+latex: \end{exampl}
 #+HTML: </details>
 
 *** Limitation                                                      :ignore:
@@ -2476,6 +2670,7 @@ Similar relationship can be found for $T$, $KS$ and $GS$.
 Determine the approximate norms of $T$, $KS$ and $GS$ for large loop gains ($|G(j\omega) K(j\omega)| \gg 1$) and small loop gains ($|G(j\omega) K(j\omega)| \ll 1$).
 
 #+HTML: <details><summary>Hint</summary>
+#+LATEX: \tcbsubtitle{Hint}
 You can follows this procedure for $T$, $KS$ and $GS$:
 1. Write the closed-loop transfer function as a function of $K(s)$ and $G(s)$
 2. Take $|K(j\omega)G(j\omega)| \gg 1$ and conclude on the norm of the closed-loop transfer function
@@ -2483,6 +2678,7 @@ You can follows this procedure for $T$, $KS$ and $GS$:
 #+HTML: </details>
 
 #+HTML: <details><summary>Answer</summary>
+#+LATEX: \tcbsubtitle{Answer}
 The obtained constrains are shown in Figure [[fig:h-infinity-4-blocs-constrains]].
 #+HTML: </details>
 #+end_exercice
@@ -2651,7 +2847,7 @@ Here is the general design procedure that will be followed:
 4. Chose the suitable weighted general plant to shape the wanted quantities
 5. Shape sequentially the chosen closed-loop transfer functions
 
-Let's first convert the system of Figure [[fig:ex_test_system]] into the classical feedback architecture of Figure [[fig:classical_feedback_test_system]].
+Let's first convert the system of Figure [[fig:ex_test_system]] into the classical feedback architecture of Figure [[fig:classical_feedback_test_system_bis]].
 
 #+begin_src latex :file ex_test_system_feedback.pdf
   \begin{tikzpicture}
@@ -2675,7 +2871,7 @@ Let's first convert the system of Figure [[fig:ex_test_system]] into the classic
   \end{tikzpicture}
 #+end_src
 
-#+name: fig:classical_feedback_test_system
+#+name: fig:classical_feedback_test_system_bis
 #+caption: Block diagram corresponding to the example system
 #+RESULTS:
 [[file:figs/ex_test_system_feedback.png]]
@@ -2819,6 +3015,7 @@ Time domain simulations will be performed by first computing the closed-loop sys
 4. Using Matlab, define the generalized plant
 
 #+HTML: <details><summary>Hint</summary>
+#+LATEX: \tcbsubtitle{Hint}
 1. Make use of Table [[tab:usual_shaping_gang_four]]
 2. Make use of Table [[tab:usual_shaping_gang_four]]
 3. See Section [[sec:shaping_multiple_tf]]
@@ -2832,13 +3029,14 @@ In such case, we want to shape $S$, $GS$ and $T$.
 
 #+name: tab:ex_specification_shapes
 #+caption: Control Specifications and associated wanted shape of the closed-loop transfer functions
-| <l>                      |               <c>               | <l>                                |
+#+attr_latex: :environment tabularx :booktabs t :width 0.7\linewidth :align llX
-| Specification            | Corresponding Transfer Function | Wanted Shape                       |
+| <l>                      |    <c>    | <l>                                |
-|--------------------------+---------------------------------+------------------------------------|
+| *Specification*          |   *TF*    | *Wanted Shape*                     |
-| Follow Step Reference    |               $S$               | +40dB of slope at low frequency    |
+|--------------------------+-----------+------------------------------------|
-| Reject Disturbances      |            $S$, $GS$            | Small gain                         |
+| Follow Step Reference    |    $S$    | +40dB of slope at low frequency    |
-| Reject measurement noise |               $T$               | Small high frequency (>100Hz) gain |
+| Reject Disturbances      | $S$, $GS$ | Small gain                         |
-| Robust System            |               $S$               | Small $\Vert S \Vert_\infty$       |
+| Reject measurement noise |    $T$    | Small high frequency (>100Hz) gain |
+| Robust System            |    $S$    | Small $\Vert S \Vert_\infty$       |
 
 To do so, we use to generalized plant shown in Figure [[fig:ex_general_plant]] for the synthesis where the three closed-loop tranfert functions from $w$ to $[z_1\,,z_2\,,z_3]$ are respectively $S$, $GS$ and $T$.
 
@@ -3504,6 +3702,8 @@ If you want to nice reference book in French, look at cite:duc99_comman_h.
 
 You can also look at the very good lectures below.
 
-yt:?listType=playlist&list=PLn8PRpmsu08qFLMfgTEzR8DxOPE7fBiin
+[[yt:?listType=playlist&list=PLn8PRpmsu08qFLMfgTEzR8DxOPE7fBiin][Robust Control - Brian Douglas]]
 
-yt:?listType=playlist&list=PLsjPUqcL7ZIFHCObUU_9xPUImZ203gB4o
+[[yt:?listType=playlist&list=PLsjPUqcL7ZIFHCObUU_9xPUImZ203gB4o][Control Bootcamp - Steve Brunton]]
+
+#+latex: \printbibliography

preamble.tex

@@ -0,0 +1,113 @@
+\usepackage{float}
+
+\usepackage{caption,tabularx,booktabs}
+
+\usepackage{biblatex}
+\addbibresource{ref.bib}
+
+\usepackage{fontawesome}
+
+\usepackage{caption}
+\usepackage{subcaption}
+
+\captionsetup[figure]{labelfont=bf}
+\captionsetup[subfigure]{labelfont=bf}
+\captionsetup[listing]{labelfont=bf}
+\captionsetup[table]{labelfont=bf}
+
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+
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+\definecolor{my-turq}{HTML}{6bc7db}
+\definecolor{my-pale-turq}{HTML}{e6f6f9}
+
+\usepackage{inconsolata}
+
+\usepackage[newfloat=true, chapter]{minted}
+\usemintedstyle{autumn}
+
+\setminted{frame=lines,breaklines=true,tabsize=4,fontsize=\scriptsize,autogobble=true,labelposition=topline,bgcolor=my-pale-grey}
+\setminted[matlab]{label=Matlab}
+\setminted[latex]{label=LaTeX}
+\setminted[bash]{label=Bash}
+\setminted[python]{label=Python}
+\setminted[text]{label=Results}
+
+\setmintedinline{fontsize=\normalsize,bgcolor=my-pale-grey}
+
+\usepackage[most]{tcolorbox}
+
+\tcbuselibrary{minted}
+
+\newtcolorbox{seealso}{   enhanced,breakable,colback=my-pale-grey,colframe=my-grey,fonttitle=\bfseries,title=See Also}
+\newtcolorbox{hint}{      enhanced,breakable,colback=my-pale-grey,colframe=my-grey,fonttitle=\bfseries,title=Hint}
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+\newtcolorbox{important}{ enhanced,breakable,colback=my-pale-red, colframe=my-red, fonttitle=\bfseries,title=Important}
+\newtcolorbox{exampl}[1][]{    enhanced,breakable,colback=my-pale-green,colframe=my-green,fonttitle=\bfseries,title=Example,#1}
+\newtcolorbox{exercice}{  enhanced,breakable,colback=my-pale-yellow,colframe=my-yellow,fonttitle=\bfseries,title=Exercice}
+\newtcolorbox{question}{  enhanced,breakable,colback=my-pale-yellow,colframe=my-yellow,fonttitle=\bfseries,title=Question}
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+
+\newtcolorbox{my-quote}[1]{%
+  colback=my-pale-grey,
+  grow to right by=-10mm,
+  grow to left by=-10mm,
+  boxrule=0pt,
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+  breakable,
+  enhanced jigsaw,
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+\renewenvironment{quote}{\begin{my-quote}}{\end{my-quote}}
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+  borderline west={4pt}{0pt}{my-grey}}
+
+\renewenvironment{verse}{\begin{my-verse}}{\end{my-verse}}
+
+\usepackage{environ}% http://ctan.org/pkg/environ
+\NewEnviron{aside}{%
+  \marginpar{\BODY}
+}
+
+\renewenvironment{verbatim}{\VerbatimEnvironment\begin{minted}[]{text}}{\end{minted}}
+
+\usepackage{soul}
+\sethlcolor{my-pale-grey}
+
+\let\OldTexttt\texttt
+\renewcommand{\texttt}[1]{{\ttfamily\hl{\mbox{\,#1\,}}}}
+
+\makeatletter
+\preto\Gin@extensions{png,}
+\DeclareGraphicsRule{.png}{pdf}{.pdf}{\noexpand\Gin@base.pdf}
+\makeatother
+
+\usepackage{hyperref}
+\hypersetup{
+  colorlinks = true,
+  allcolors = my-blue
+}
+
+\usepackage{hypcap}

ref.bib

@@ -0,0 +1,33 @@
+@article{lurie02_system_archit_trades_using_bode,
+  author = {Boris J. Lurie and Ali Ghavimi and Fred Y. Hadaegh and Edward Mettler},
+  title = {System Architecture Trades Using Bode-Step Control Design},
+  journal = {Journal of Guidance, Control, and Dynamics},
+  volume = {25},
+  number = {2},
+  pages = {309-315},
+  year = {2002},
+  doi = {10.2514/2.4883},
+  url = {https://doi.org/10.2514/2.4883},
+}
+@techreport{bibel92_guidel_h,
+  author = {Bibel, John E and Malyevac, D Stephen},
+  institution = {NAVAL SURFACE WARFARE CENTER DAHLGREN DIV VA},
+  keywords = {robust control},
+  title = {Guidelines for the selection of weighting functions for H-infinity control},
+  year = {1992},
+}
+@book{skogestad07_multiv_feedb_contr,
+  author = {Skogestad, Sigurd and Postlethwaite, Ian},
+  title = {Multivariable Feedback Control: Analysis and Design},
+  year = {2007},
+  publisher = {John Wiley},
+  isbn = {9780470011683},
+  keywords = {favorite},
+}
+@book{duc99_comman_h,
+  author = {Duc, G and Font, S},
+  title = {Commande H infinie et mu-analyse-des outils pour la robustesse},
+  year = {1999},
+  publisher = {Lavoisier},
+  journal = {Hermes., Paris: Hermes},
+}