Add midterm LaTeX source and PDF

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diff --git a/Midterm_19-150-903.pdf b/Midterm_19-150-903.pdf
new file mode 100644
index 0000000..d8b3424
--- /dev/null
+++ b/Midterm_19-150-903.pdf
diff --git a/RCCO.bib b/RCCO.bib
new file mode 100644
index 0000000..56aba71
--- /dev/null
+++ b/RCCO.bib
@@ -0,0 +1,207 @@
+
+@article{pflimlin_modeling_2010,
+	title = {Modeling and attitude control analysis of a ducted-fan micro aerial vehicle},
+	volume = {18},
+	issn = {09670661},
+	url = {https://linkinghub.elsevier.com/retrieve/pii/S0967066109001828},
+	doi = {10.1016/j.conengprac.2009.09.009},
+	pages = {209--218},
+	number = {3},
+	journaltitle = {Control Engineering Practice},
+	shortjournal = {Control Engineering Practice},
+	author = {Pflimlin, J.-M. and Binetti, P. and Souères, P. and Hamel, T. and Trouchet, D.},
+	urldate = {2024-03-25},
+	date = {2010-03},
+	langid = {english},
+	file = {Pflimlin et al. - 2010 - Modeling and attitude control analysis of a ducted.pdf:/Users/npross/Zotero/storage/NYXNQRF2/Pflimlin et al. - 2010 - Modeling and attitude control analysis of a ducted.pdf:application/pdf},
+}
+
+@inproceedings{pflimlin_hovering_2004,
+	location = {Nassau, Bahamas},
+	title = {Hovering flight stabilization in wind gusts for ducted fan {UAV}},
+	isbn = {978-0-7803-8682-2},
+	url = {http://ieeexplore.ieee.org/document/1429251/},
+	doi = {10.1109/CDC.2004.1429251},
+	eventtitle = {2004 43rd {IEEE} Conference on Decision and Control ({CDC}) ({IEEE} Cat. No.04CH37601)},
+	pages = {3491--3496 Vol.4},
+	booktitle = {2004 43rd {IEEE} Conference on Decision and Control ({CDC}) ({IEEE} Cat. No.04CH37601)},
+	publisher = {{IEEE}},
+	author = {Pflimlin, J.M. and Soueres, P. and Hamel, T.},
+	urldate = {2024-03-25},
+	date = {2004},
+	file = {Pflimlin et al. - 2004 - Hovering flight stabilization in wind gusts for du.pdf:/Users/npross/Zotero/storage/K5XIP2TS/Pflimlin et al. - 2004 - Hovering flight stabilization in wind gusts for du.pdf:application/pdf},
+}
+
+@inproceedings{pflimlin_aerodynamic_2007,
+	location = {Kos, Greece},
+	title = {Aerodynamic modeling and practical attitude stabilization of a ducted fan {UAV}},
+	isbn = {978-3-9524173-8-6},
+	url = {https://ieeexplore.ieee.org/document/7068497/},
+	doi = {10.23919/ECC.2007.7068497},
+	eventtitle = {2007 European Control Conference ({ECC})},
+	pages = {4023--4029},
+	booktitle = {2007 European Control Conference ({ECC})},
+	publisher = {{IEEE}},
+	author = {Pflimlin, Jean Michel and Binetti, Paolo and Trouchet, Daniel and Soueres, Philippe and Hamel, Tarek},
+	urldate = {2024-03-25},
+	date = {2007-07},
+}
+
+@article{naldi_design_2010,
+	title = {Design and experimental validation of a nonlinear control law for a ducted-fan miniature aerial vehicle},
+	volume = {18},
+	issn = {09670661},
+	url = {https://linkinghub.elsevier.com/retrieve/pii/S0967066110000444},
+	doi = {10.1016/j.conengprac.2010.02.007},
+	pages = {747--760},
+	number = {7},
+	journaltitle = {Control Engineering Practice},
+	shortjournal = {Control Engineering Practice},
+	author = {Naldi, Roberto and Gentili, Luca and Marconi, Lorenzo and Sala, Andrea},
+	date = {2010-07},
+	langid = {english},
+	file = {Naldi et al. - 2010 - Design and experimental validation of a nonlinear .pdf:/Users/npross/Zotero/storage/PINX95Y4/Naldi et al. - 2010 - Design and experimental validation of a nonlinear .pdf:application/pdf},
+}
+
+@inproceedings{ko_ducted_2007,
+	location = {Hilton Head, South Carolina},
+	title = {Ducted Fan {UAV} Modeling and Simulation in Preliminary Design},
+	isbn = {978-1-62410-160-1},
+	url = {https://arc.aiaa.org/doi/10.2514/6.2007-6375},
+	doi = {10.2514/6.2007-6375},
+	eventtitle = {{AIAA} Modeling and Simulation Technologies Conference and Exhibit},
+	booktitle = {{AIAA} Modeling and Simulation Technologies Conference and Exhibit},
+	publisher = {American Institute of Aeronautics and Astronautics},
+	author = {Ko, Andy and Ohanian, Osgar and Gelhausen, Paul},
+	date = {2007-08-20},
+	langid = {english},
+	file = {Ko et al. - 2007 - Ducted Fan UAV Modeling and Simulation in Prelimin.pdf:/Users/npross/Zotero/storage/9VB5WUZT/Ko et al. - 2007 - Ducted Fan UAV Modeling and Simulation in Prelimin.pdf:application/pdf},
+}
+
+@thesis{rothlin_flight_2021,
+	location = {Rapperswil, Switzerland},
+	title = {Flight control system for hovering a 'Ducted-Fan' prototype},
+	url = {https://www.ost.ch/de/details/abstracts/ducted-fan-prototype-17443},
+	pagetotal = {114},
+	institution = {{OST} {RJ} University of Applied Sciences},
+	type = {Bachelor Thesis},
+	author = {Rothlin, Tobias and Tönz, Tim},
+	date = {2021},
+	file = {Rothlin and Tönz - 2021 - Flight control system for hovering a 'Ducted-Fan' .pdf:/Users/npross/Zotero/storage/6E434P4J/Rothlin and Tönz - 2021 - Flight control system for hovering a 'Ducted-Fan' .pdf:application/pdf},
+}
+
+@inproceedings{koo_output_1998,
+	location = {Tampa, {FL}, {USA}},
+	title = {Output tracking control design of a helicopter model based on approximate linearization},
+	volume = {4},
+	isbn = {978-0-7803-4394-8},
+	url = {http://ieeexplore.ieee.org/document/761745/},
+	doi = {10.1109/CDC.1998.761745},
+	eventtitle = {37th {IEEE} Conference on Decision and Control},
+	pages = {3635--3640},
+	booktitle = {Proceedings of the 37th {IEEE} Conference on Decision and Control (Cat. No.98CH36171)},
+	publisher = {{IEEE}},
+	author = {Koo, T.J. and Sastry, S.},
+	date = {1998},
+	file = {Koo and Sastry - 1998 - Output tracking control design of a helicopter mod.pdf:/Users/npross/Zotero/storage/KDPUPDMP/Koo and Sastry - 1998 - Output tracking control design of a helicopter mod.pdf:application/pdf},
+}
+
+@book{stengel_flight_2022,
+	location = {Princeton Oxford},
+	edition = {Second edition},
+	title = {Flight dynamics},
+	isbn = {978-0-691-22025-3},
+	pagetotal = {894},
+	publisher = {Princeton University Press},
+	author = {Stengel, Robert F.},
+	date = {2022},
+	file = {Table of Contents PDF:/Users/npross/Zotero/storage/2TBQDVR6/Stengel - 2022 - Flight dynamics.pdf:application/pdf},
+}
+
+@article{packard_complex_1993,
+	title = {The complex structured singular value},
+	volume = {29},
+	rights = {https://www.elsevier.com/tdm/userlicense/1.0/},
+	issn = {00051098},
+	url = {https://linkinghub.elsevier.com/retrieve/pii/000510989390175S},
+	doi = {10.1016/0005-1098(93)90175-S},
+	pages = {71--109},
+	number = {1},
+	journaltitle = {Automatica},
+	shortjournal = {Automatica},
+	author = {Packard, A. and Doyle, J.},
+	date = {1993-01},
+	langid = {english},
+	file = {Accepted Version:/Users/npross/Zotero/storage/F6DIYFGM/Packard and Doyle - 1993 - The complex structured singular value.pdf:application/pdf},
+}
+
+@article{martini_dynamic_2009,
+	title = {Dynamic Modelling and Stability Analysis of Model-Scale Helicopters Under Wind Gust},
+	volume = {54},
+	rights = {http://www.springer.com/tdm},
+	issn = {0921-0296, 1573-0409},
+	url = {http://link.springer.com/10.1007/s10846-008-9280-z},
+	doi = {10.1007/s10846-008-9280-z},
+	pages = {647--686},
+	number = {4},
+	journaltitle = {Journal of Intelligent and Robotic Systems},
+	shortjournal = {J Intell Robot Syst},
+	author = {Martini, Adnan and Léonard, François and Abba, Gabriel},
+	date = {2009-04},
+	langid = {english},
+	file = {Martini et al. - 2009 - Dynamic Modelling and Stability Analysis of Model-.pdf:/Users/npross/Zotero/storage/2SPK4W3M/Martini et al. - 2009 - Dynamic Modelling and Stability Analysis of Model-.pdf:application/pdf},
+}
+
+@article{chirarattananon_dynamics_2017,
+	title = {Dynamics and flight control of a flapping-wing robotic insect in the presence of wind gusts},
+	volume = {7},
+	issn = {2042-8898, 2042-8901},
+	url = {https://royalsocietypublishing.org/doi/10.1098/rsfs.2016.0080},
+	doi = {10.1098/rsfs.2016.0080},
+	abstract = {With the goal of operating a biologically inspired robot autonomously outside of laboratory conditions, in this paper, we simulated wind disturbances in a laboratory setting and investigated the effects of gusts on the flight dynamics of a millimetre-scale flapping-wing robot. Simplified models describing the disturbance effects on the robot's dynamics are proposed, together with two disturbance rejection schemes capable of estimating and compensating for the disturbances. The proposed methods are experimentally verified. The results show that these strategies reduced the root-mean-square position errors by more than 50\% when the robot was subject to 80 cm s
+              −1
+              horizontal wind. The analysis of flight data suggests that modulation of wing kinematics to stabilize the flight in the presence of wind gusts may indirectly contribute an additional stabilizing effect, reducing the time-averaged aerodynamic drag experienced by the robot. A benchtop experiment was performed to provide further support for this observed phenomenon.},
+	pages = {20160080},
+	number = {1},
+	journaltitle = {Interface Focus},
+	shortjournal = {Interface Focus.},
+	author = {Chirarattananon, Pakpong and Chen, Yufeng and Helbling, E. Farrell and Ma, Kevin Y. and Cheng, Richard and Wood, Robert J.},
+	date = {2017-02-06},
+	langid = {english},
+	file = {Full Text:/Users/npross/Zotero/storage/UKKFL5SQ/Chirarattananon et al. - 2017 - Dynamics and flight control of a flapping-wing rob.pdf:application/pdf},
+}
+
+@thesis{pereira_hover_2008,
+	title = {Hover and Wind-Tunnel Testing of Shrouded Rotors for Improved Micro Air Vehicle Design},
+	url = {http://hdl.handle.net/1903/8752},
+	abstract = {This dissertation describes an experimental investigation of the effects of varying the shroud profile shape on the performance of {MAV} (Micro Air Vehicle) -scale shrouded rotors. Hover tests were performed on seventeen models with a rotor diameter of 16 cm (6.3 in) and various values of diffuser expansion angle, diffuser length, inlet lip radius and blade tip clearance, at various rotor collective angles. Compared to the baseline open rotor, the shrouded rotors showed increases in thrust by up to 94\%, at the same power consumption, or reductions in power by up to 62\% at the same thrust. These improvements surpass those predicted by momentum theory, due to the additional effect of the shrouds in reducing the non-ideal power losses of the rotor. The uniformity of the rotor wake was improved by the presence of the shrouds and by decreasing the blade tip clearance, resulting in lower induced power losses. Strong suction pressures were observed on the shroud inlet surface, at the blade passage region; taking advantage of this phenomenon could enable further increases in thrust. However, trade studies showed that, for a given overall aircraft size limitation, and ignoring considerations of the safety benefits of a shroud, a larger-diameter open rotor is more likely to give better performance than a smaller-diameter shrouded rotor. A single shrouded-rotor model was subsequently tested in translational flight at various angles of attack. In axial flow, at the same collective, the net thrust and the power consumption of the shrouded rotor were lower than those of the open rotor; in edgewise flow, the shrouded rotor produced greater thrust than the open rotor, while consuming less power. Measurements of the shroud surface pressure distributions illustrated the extreme asymmetry of the flow around the shroud, with consequent pitch moments much greater than those experienced by the open rotor. Except at low airspeeds and high angles of attack, the static pressure in the wake did not reach ambient atmospheric values at the diffuser exit plane; this challenges the validity of the fundamental assumption of the simple-momentum-theory flow model for short-chord shrouds in translational flight.},
+	institution = {University of Maryland (College Park, Md.)},
+	type = {phdthesis},
+	author = {Pereira, Jason Louie},
+	urldate = {2024-03-31},
+	date = {2008-08-29},
+	langid = {american},
+	file = {Full Text PDF:/Users/npross/Zotero/storage/L7WYTRQK/Pereira - 2008 - Hover and Wind-Tunnel Testing of Shrouded Rotors f.pdf:application/pdf},
+}
+
+@book{skogestad_multivariable_2010,
+	location = {Chichester},
+	edition = {2. ed., repr},
+	title = {Multivariable feedback control: analysis and design},
+	isbn = {978-0-470-01168-3 978-0-470-01167-6},
+	shorttitle = {Multivariable feedback control},
+	pagetotal = {574},
+	publisher = {Wiley},
+	author = {Skogestad, Sigurd and Postlethwaite, Ian},
+	date = {2010},
+}
+
+@thesis{rothlin_optimization_2024,
+	location = {Rapperswil, Switzerland},
+	title = {Optimization of a ducted fan prototype},
+	url = {https://www.ost.ch/de/details/abstracts/optimization-of-a-ducted-fan-prototype-18926},
+	institution = {{OST} {RJ} University of Applied Sciences},
+	type = {Semester Project},
+	author = {Rothlin, Tobias},
+	date = {2024},
+}
diff --git a/fig/generalized_plant.tex b/fig/generalized_plant.tex
new file mode 100644
index 0000000..cbc3bf9
--- /dev/null
+++ b/fig/generalized_plant.tex
@@ -0,0 +1,40 @@
+\begin{tikzpicture}[
+      plant/.style = {
+        draw, thick,
+        minimum width = 1cm,
+        minimum height = 1cm,
+      },
+      arrow/.style = {
+        -latex, thick,
+      },
+    ]
+  \matrix [row sep = 4mm, nodes={plant}] {
+    \node [fill=red!10] (U) {$\Delta$}; \\
+    \node [fill=blue!10] (G) {$G$}; \\ 
+    \node [fill=green!10] (K) {$K$}; \\
+  };
+
+  \coordinate (Uz) at (U.west);
+  \coordinate (Uv) at (U.east);
+
+  \coordinate (Gz) at ($(G.north west) - (0, 2mm)$);
+  \coordinate (Ge) at (G.west);
+  \coordinate (Gy) at ($(G.south west) + (0, 2mm)$);
+
+  \coordinate (Gv) at ($(G.north east) - (0, 2mm)$);
+  \coordinate (Gw) at (G.east);
+  \coordinate (Gu) at ($(G.south east) + (0, 2mm)$);
+  
+  \coordinate (Ky) at (K.west);
+  \coordinate (Ku) at (K.east);
+
+  \begin{scope}[arrow]
+    \draw (Gz) -- ++(-1cm, 0) |- (Uz) node[left, pos=.25] {$z$};
+    \draw (Ge) -- ++(-1cm, 0) -- ++(-5mm, 0) node[left] {$e$};
+    \draw (Gy) -- ++(-1cm, 0) |- (Ky) node[left, pos=.25] {$y$};
+    
+    \draw (Uv) -- ++(1cm, 0) |- (Gv) node[right, pos=.25] {$v$};
+    \draw (Gw) ++(15mm, 0) node[right] {$w$} -- (Gw);
+    \draw (Ku) -- ++(1cm, 0) |- (Gu) node[right, pos=.25] {$u$};
+  \end{scope}
+\end{tikzpicture}
diff --git a/fig/sketch.tex b/fig/sketch.tex
new file mode 100644
index 0000000..e1eae4a
--- /dev/null
+++ b/fig/sketch.tex
@@ -0,0 +1,195 @@
+\begin{tikzpicture}[3d view={120}{15}]
+  \newcommand\radius{15mm}
+  \newcommand\height{3cm}
+
+  \begin{scope}[canvas is yx plane at z=.8]
+    \foreach \i in {1,2,3,...,8} {
+      \fill[lightgray, opacity=.5] (0,0) -- ({\i*360/8}:{.9*\radius}) 
+      arc ({\i*360/8}:{(\i+.5)*360/8}:{.9*\radius}) -- cycle;
+    };
+    \fill[lightgray, opacity=.5] (0,0) circle (3mm);
+
+    \draw[black, <-] (-1, 1) to[out=180, in=0] ++(-.5,1.2)
+      node[left, anchor = east] {Propeller};
+  \end{scope}
+
+  \begin{scope}[canvas is yx plane at z=-2]
+    \draw[black, <-] (-1, .5) to[out=180, in=0] ++(.1,1.5)
+      node[left, anchor = east] {Flap};
+  \end{scope}
+
+  \begin{scope}[canvas is xz plane at y=0, thick]
+    % flap along y back
+    \fill[lightgray!30, draw=lightgray] ({-.3*\radius}, -2) rectangle ({-.8*\radius}, -2.5);
+
+    % drag force of flap along y back
+    \draw[-latex, blue!80!black] ({-.55*\radius}, -2) -- ++(0,-6mm);
+  \end{scope}
+
+  % FIXME: why can't is use -.55\radius??
+  \begin{scope}[canvas is yz plane at x=-.825, thick]
+    % lift force of flap along y back
+    \draw[-latex, green!50!black] (0, -2) -- ++(5mm,0);
+  \end{scope}
+
+  \begin{scope}[canvas is yz plane at x=0, thick]
+    % flap along x front
+    \fill[lightgray!30, draw=lightgray] ({.3*\radius}, -2) rectangle ({.8*\radius}, -2.5);
+    % drag force of flap along x front
+    \draw[-latex, blue!80!black] ({.55*\radius}, -2) -- ++(0,-6mm)
+      node[right] {$\vec{F}_{d,i}$};
+
+    % flap along x back
+    \fill[lightgray!30, draw=lightgray] ({-.3*\radius}, -2) rectangle ({-.8*\radius}, -2.5);
+    % drag force of flap along x back 
+    \draw[-latex, blue!80!black] ({-.55*\radius}, -2) -- ++(0,-6mm);
+  \end{scope}
+
+  % FIXME: why can't is use .55\radius??
+  \begin{scope}[canvas is xz plane at y=-.825, thick]
+    % lift force of flap along x back
+    \draw[-latex, green!50!black] (0, -2) -- ++(5mm,0);
+  \end{scope}
+
+  % FIXME: why can't is use .55\radius??
+  \begin{scope}[canvas is xz plane at y=.825, thick]
+    % lift force of flap along x front
+    \draw[-latex, green!50!black] (0, -2) -- ++(5mm,0);
+
+    % flap angle for front x
+    \fill[violet] (0,-2) circle (.5mm);
+    \node[violet, right, yshift=1mm] at (0,-2) {$\alpha_1$};
+  \end{scope}
+
+  \begin{scope}[canvas is xz plane at y=0, thick]
+    % flap along y front
+    \fill[lightgray!30, draw=lightgray] ({.3*\radius}, -2) rectangle ({.8*\radius}, -2.5);
+
+    % drag force of flap along y front
+    \draw[-latex, blue!80!black] ({.55*\radius}, -2) -- ++(0,-6mm);
+  \end{scope}
+
+  \begin{scope}[canvas is xy plane at z=0, thick]
+    \draw[lightgray, dashed] (0,0) circle (\radius);
+  \end{scope}
+
+  \begin{scope}[canvas is xy plane at z=-2, thick]
+    % lower circle
+    \draw[gray] (0,0) circle (\radius);
+    \draw[lightgray, densely dotted] (-\radius, 0) -- (\radius, 0);
+    \draw[lightgray, densely dotted] (0, -\radius) -- (0, \radius);
+  \end{scope}
+
+  \begin{scope}[canvas is xy plane at z=1, thick]
+    % upper circle
+    \draw[gray] (0,0) circle (\radius);
+    \draw[lightgray, densely dotted] (-\radius, 0) -- (\radius, 0);
+    \draw[lightgray, densely dotted] (0, -\radius) -- (0, \radius);
+  \end{scope}
+
+  % FIXME: why can't is use .55\radius??
+  \begin{scope}[canvas is yz plane at x=.825, thick]
+    % lift force of flap along y front
+    \draw[-latex, green!50!black] (0, -2) -- ++(5mm,0)
+      node[pos=1, below, yshift=-1mm] {$\vec{F}_{\ell,i}$};
+  \end{scope}
+
+  \begin{scope}[canvas is yz plane at x=-.825, thick]
+    % flap angle for back y
+    \fill[violet] (0,-2) circle (.5mm);
+    \node[violet, xshift=2.5mm, yshift=1.5mm] at (0,-2) {$\alpha_4$};
+  \end{scope}
+
+  \begin{scope}[canvas is xz plane at y=-.825, thick]
+    % flap angle for back x
+    \fill[violet] (0,-2) circle (.5mm);
+    \node[violet, xshift=2.5mm, yshift=1.5mm] at (0,-2) {$\alpha_3$};
+  \end{scope}
+
+  \begin{scope}[canvas is yz plane at x=.825, thick]
+    % lift force of flap along y front
+    \fill[violet] (0,-2) circle (.5mm);
+    \node[violet, xshift=2mm, yshift=1.5mm] at (0,-2) {$\alpha_2$};
+  \end{scope}
+
+  \begin{scope}[thick]
+    % cylinder border
+    \draw[gray] (xyz cylindrical cs:z=-2,radius=1.5,angle=-60)
+      -- ++ (xyz cylindrical cs:z=3);
+
+    \draw[gray] (xyz cylindrical cs:z=-2,radius=1.5,angle=120)
+      -- ++ (xyz cylindrical cs:z=3);
+
+    % central axis
+    \draw[lightgray, dashdotted]
+      (xyz cylindrical cs:z=-2) -- (xyz cylindrical cs:z=2);
+
+    % outer line for radius measurement
+    \coordinate (TO) at (xyz cylindrical cs:z=1);
+    \coordinate (TR) at (xyz cylindrical cs:z=1,radius=1.5,angle=120);
+    \draw[red!60!black, <->] (TO) -- node[midway, above] {$a$} (TR);
+  \end{scope}
+
+  \begin{scope}[canvas is yz plane at x=0, thick]
+    % top line for center of mass measurement
+    \draw[lightgray, densely dotted] (0, 0) -- (2.5, 0);
+    % bottom line for center of mass measurement
+    \draw[lightgray, densely dotted] (\radius, -2) -- (2.5, -2);
+
+    % Center of mass height measurement
+    \draw[red!60!black, <->] (2.5,0) -- node[midway, right] {$d$} (2.5, -2);
+
+    % thrust force
+    \draw[blue!80!black, -latex] (0,1) -- node[near end, right] {$\vec{F}_{T}$} (0,2);
+  \end{scope}
+
+  \begin{scope}[canvas is xy plane at z=1.25, thick]
+    % propeller angular velocity
+    \draw[violet, ->] (290:.2) arc[start angle=290, end angle=-30, radius=.2]
+      node[pos=0, left] {$\omega$};
+  \end{scope}
+
+  \begin{scope}[canvas is xz plane at y=0, thick]
+    % line to pitch angle
+    \draw[lightgray, densely dotted] (0, 0) -- (3, 0);
+  \end{scope}
+
+  \begin{scope}[canvas is xz plane at y=1.85, thick]
+    % roll angle
+    \draw[violet, ->] (-40:.2) arc[start angle=-40, end angle=-370, radius=.2]
+      node[midway, right] {$\phi$};
+  \end{scope}
+
+  \begin{scope}[canvas is yz plane at x=2.5, thick]
+    % pitch angle
+    \draw[violet, ->] (40:.2) arc[start angle=40, end angle=370, radius=.2]
+      node[midway, above left] {$\vartheta$};
+  \end{scope}
+
+  \begin{scope}[canvas is xy plane at z=-1.25, thick]
+    % yaw angle
+    \draw[violet, ->] (290:.2) arc[start angle=290, end angle=-30, radius=.2]
+      node[midway, right] {$\psi$};
+  \end{scope}
+
+  \begin{scope}[thick]
+    % body coordinate system
+    \coordinate(BO) at (0,0,0);
+    \draw[-latex] (BO) -- (xyz cylindrical cs:radius=.5)
+      node[left] {$\uvec{y}$};
+    \draw[-latex] (BO) -- (xyz cylindrical cs:radius=.5,angle=90)
+      node[right] {$\uvec{x}$};
+    \draw[-latex] (BO) -- (xyz cylindrical cs:z=-.5)
+      node[below] {$\uvec{z}$};
+
+    % intertial coordinate system
+    \coordinate (IO) at (xyz cylindrical cs:radius=3.5,angle=-70,z=-1.5);
+
+    \draw[-latex] (IO) -- ++ (xyz cylindrical cs:radius=.5)
+      node[left] {$\uvec{\jmath}$};
+    \draw[-latex] (IO) -- ++ (xyz cylindrical cs:radius=.5,angle=90)
+      node[right] {$\uvec{\imath}$};
+    \draw[-latex] (IO) -- ++ (xyz cylindrical cs:z=-.5)
+      node[below] {$\uvec{k}$};
+  \end{scope}
+\end{tikzpicture}
diff --git a/fig/sketch_hand.pdf b/fig/sketch_hand.pdf
new file mode 100644
index 0000000..afbbd69
--- /dev/null
+++ b/fig/sketch_hand.pdf
diff --git a/fig/stepsim.dat b/fig/stepsim.dat
new file mode 100644
index 0000000..2a3f1c1
--- /dev/null
+++ b/fig/stepsim.dat
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diff --git a/midterm.tex b/midterm.tex
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--- /dev/null
+++ b/midterm.tex
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+\documentclass[a4paper, 10pt]{IEEEtran}
+
+\usepackage{array}
+\usepackage{booktabs}
+\usepackage{siunitx}
+\usepackage{float}
+
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{oststud}
+
+\usepackage{graphicx}
+\usepackage{subcaption}
+\usepackage{pgfplots}
+\pgfplotsset{compat=1.18}
+\usepackage{tikz}
+\usetikzlibrary{3d,perspective,matrix,calc}
+
+% Bibliography IEEE style
+\usepackage[
+  backend=biber,
+  style=ieee,
+  citestyle=numeric-comp,
+  sortcites=true
+]{biblatex}
+
+% Allow breaking long URLs in the bibliography
+\setcounter{biburllcpenalty}{7000}
+\setcounter{biburlucpenalty}{8000}
+
+% Add a bibliography resource
+\addbibresource{RCCO.bib}
+
+\title{
+  Robust $\mathcal{H}_\infty$ Controller Synthesis for Hovering Ducted-Fan VTOL
+  Micro-UAV
+}
+\author{
+  \IEEEauthorblockN{Naoki Sean Pross} \IEEEauthorrefmark{1}
+  % \thanks{
+  % }
+  \thanks{
+    Midterm project for course \emph{Robust Control and Convex Optimization}
+    taught by Prof. Dr. R. S. Smith at ETH Zurich.
+    \IEEEauthorrefmark{1}Student ID:~\texttt{19-150-903}
+    Email:~\texttt{npross@student.ethz.ch}.
+    Thanks to T. Rothlin for providing the electromechanical parameters for the
+    UAV.
+  }
+}
+
+\begin{document}
+
+\maketitle
+
+\begin{abstract}
+  This paper models an experimental vertical take-off landing (VTOL) micro
+  unmanned aerial vehicle (UAV) device developed at the University of Applied
+  Sciences OST RJ in Rapperswil. The model is then converted to a linear time
+  invariant (LTI) state space plant to design a robust controller using
+  $\mathcal{H}_\infty$ synthesis that tracks a piecewise constant spatial
+  position reference. In conclusion simulated preliminary results on the
+  controller performance are presented.
+\end{abstract}
+
+\section{Introduction}
+
+The presented VTOL micro UAV is a device developed by T. Rothlin
+\cite{rothlin_optimization_2024}  at the University of Applied Sciences OST RJ,
+Rapperswil that can be steered only by controlling the propeller velocity and
+the angle of attack of four flaps under the ducted fan. Currently there is not
+a control algorithm for the UAV and the uncertainties from aerodynamics and
+gyroscopic effects caused by the propeller make this an interesting candidate
+for robust controller synthesis.
+
+% \subsection{Notation}
+% 
+% Throughout this text we will make use a notation customary to mechanics, using
+% boldface letters for vectors and matrices and a hat to indicate unit vectors.
+% Spatial coordinates are grouped into a position vector $\vec{P}$ to avoid
+% confusing the $x$ and $y$ coordinates with the state space model's output and
+% state vectors $\vec{y}$ and $\vec{x}$.
+
+% \begin{table}
+%   \centering
+%   \caption{
+%     Summary of notation for physical quantities.
+%     \label{tab:sym-phy}
+%   }
+%   \begin{tabular}{cl}
+%     \toprule
+%     Symbol & Physical Quantity \\
+%     \midrule
+%     $\vec{P}$ & Position vector \\
+%     $\vec{F}$ & Total force \\
+%     $\vec{M}$ & Total torque \\
+%     \bottomrule
+%   \end{tabular}
+% \end{table}
+
+% \subsection{Organization}
+% 
+% Section \ref{sec:modelling} presents a derivation of a model for the plant, and
+% then \S\ref{sec:control} illustrates the controller design. Finally, in
+% \S\ref{sec:simulation} the closed loop design is simulated and briefly
+% discussed.
+
+\section{System Modelling}
+\label{sec:modelling}
+
+% \subsection{Reference Frames}
+
+To model the dynamics of the ducted-fan UAV two reference frames are required:
+an inertial frame and a body-frame attached to the center of mass
+\cite{stengel_flight_2022}. In the inertial frame we work in the base given by
+the unit vectors $\{\uvec{\imath},\uvec{\jmath},\uvec{k}\}$, whereas in the
+body-frame we use $\{\uvec{x}, \uvec{y}, \uvec{z}\}$ together with the Euler
+angles $\vec{\Theta} = [ \phi ~ \vartheta ~ \psi ]^\mathsf{T}$. The angular
+velocity $\vec{\Omega} = [p ~ q ~ r]^\mathsf{T}$ is then related to
+$\vec{\Theta}$ by $\dot{\vec{\Theta}} = \mx{U} \vec{\Omega}$. To rotate from
+the inertial frame to body frame we use the $SO(3)$ matrix $\mx{R}$. Both
+$\mx{U}$ and $\mx{R}$ can be found in \cite{stengel_flight_2022}.
+
+% \begin{equation}
+%   \dot{\vec{\Theta}} = \begin{bmatrix}
+%     1 & S_\phi T_\vartheta & C_\phi T_\vartheta \\
+%     0 & C_\phi & - S_\phi \\
+%     0 & S_\phi / C_\vartheta & C_\phi / C_\vartheta
+%   \end{bmatrix}
+%   \vec{\Omega}
+%   = \mx{u} \vec{\omega},
+% \end{equation}
+% wherein we use $S_\alpha$, $C_\alpha$ and $T_\alpha$ as shorthand for $\sin
+% \alpha$, $\cos \alpha$ and $\tan \alpha$ respectively. To rotate from the
+% inertial frame to body frame we use the $SO(3)$ matrix
+% \begin{equation*}
+%   \mx{R} = \begin{bmatrix}
+%     C_\vartheta C_\psi & C_\vartheta S_\psi & - S_\vartheta \\
+%     S_\phi S_\vartheta C_\psi - C_\phi S_\psi 
+%       & S_\phi S_\vartheta C_\psi + C_\phi C_\psi
+%       & S_\phi C_\vartheta \\
+%     C_\phi S_\theta C_\psi + S_\phi S_\psi
+%       & C_\phi S_\theta S_\psi - S_\phi C_\psi
+%       & C_\phi C_\vartheta
+%   \end{bmatrix}.
+% \end{equation*}
+
+\subsection{Equations of Motion}
+
+Consider a simplified rigid-body physical model sketched in
+Fig.~\ref{fig:phy-sketch}. With respect to the inertial frame the equations of
+motion from the Newton-Euler formalism are
+\begin{subequations}
+  \begin{align}
+    m \ddot{\vec{P}} &= \mt{\mx{R}} \vec{F} \\
+    \vec{J} \dot{\vec{\Omega}} &= 
+      - \vec{\Omega} \crossp \vec{J} \vec{\Omega} + \vec{\tau},
+  \end{align}
+\end{subequations}
+where $m$ is the total mass, $\vec{P}$ is the position, $\vec{F}$ and
+$\vec{\tau}$ the total force and torque in the body frame respectively, and
+finally $\mx{J}$ the moment of inertia, which is assumed to be a diagonal
+matrix.
+
+\begin{figure}
+  \centering
+  % \includegraphics[width=.7\linewidth, trim={0 10 0 0}, clip]{fig/sketch_hand}
+  \input{fig/sketch}
+  \caption{
+    Sketch of the rigid body physical model for the UAV.
+    % \texttt{TODO: replace with computer-drawn sketch.}
+    \label{fig:phy-sketch}
+  }
+\end{figure}
+
+\begin{figure*}[t]
+  \centering
+  \begin{tikzpicture}[font = \footnotesize]
+    \begin{axis}[
+        width = 42mm, height = 28mm,
+        scale only axis,
+        grid = major,
+        grid style = {densely dotted, gray!50},
+        ylabel = {Flap Angle $\alpha$ / degrees},
+        xlabel = {Time $t$ / seconds},
+        legend entries = {
+          $\alpha_1(t)$,
+          $\alpha_2(t)$,
+          $\alpha_3(t)$,
+          $\alpha_4(t)$
+        },
+        legend columns = 2,
+        title = {\scshape Step Response of Flap Angles},
+        /tikz/line join=bevel,
+      ]
+      \addplot+ [mark=none] table [x index = 0, y index = 7] {fig/stepsim.dat};
+      \addplot+ [mark=none] table [x index = 0, y index = 8] {fig/stepsim.dat};
+      \addplot+ [mark=none] table [x index = 0, y index = 9] {fig/stepsim.dat};
+      \addplot+ [mark=none] table [x index = 0, y index = 10] {fig/stepsim.dat};
+    \end{axis}
+  \end{tikzpicture}
+  \hfill
+  \begin{tikzpicture}[font = \footnotesize, ]
+    \begin{axis}[
+        width = 42mm, height = 28mm,
+        scale only axis,
+        grid = major,
+        grid style = {densely dotted, gray!50},
+        ylabel = {Euler Angle / degrees},
+        xlabel = {Time $t$ / seconds},
+        legend entries = {
+          $\phi(t)$,
+          $\vartheta(t)$,
+          $\psi(t)$,
+        },
+        legend columns = 2,
+        legend style = {anchor = south east, at = {(.95,.05)}},
+        title = {\scshape Step Response of Euler Angles},
+      ]
+      \addplot+ [thick, mark=none] table [x index = 0, y index = 11] {fig/stepsim.dat};
+      \addplot+ [thick, mark=none] table [x index = 0, y index = 12] {fig/stepsim.dat};
+      \addplot+ [thick, mark=none] table [x index = 0, y index = 13] {fig/stepsim.dat};
+    \end{axis}
+  \end{tikzpicture}
+  \hfill
+  \begin{tikzpicture}[font = \footnotesize]
+    \pgfplotsset{set layers}
+    \begin{axis}[
+        width = 42mm, height = 28mm,
+        scale only axis,
+        grid = major,
+        grid style = {densely dotted, gray!50},
+        axis y line* = left,
+        ylabel = {Coordinate / meters},
+        xlabel = {Time $t$ / seconds},
+        legend entries = {
+          $x(t)$,
+          $y(t)$
+        },
+        title = {\scshape Step Response of Position},
+      ]
+      \addplot+ [thick, mark=none] table [x index = 0, y index = 1] {fig/stepsim.dat};
+      \addplot+ [thick, mark=none] table [x index = 0, y index = 2] {fig/stepsim.dat};
+    \end{axis}
+    \begin{axis}[
+        width = 42mm, height = 28mm,
+        scale only axis,
+        % grid = major,
+        % grid style = {densely dotted, gray!50},
+        axis y line* = right,
+        axis x line = none,
+        ylabel = {Velocity / \unit{\meter\per\second}},
+      ]
+      \addplot+ [thick, mark=none, densely dotted, opacity=.8]
+        table [x index = 0, y index = 4] {fig/stepsim.dat};
+      \addplot+ [thick, mark=none, densely dotted, opacity=.8]
+        table [x index = 0, y index = 5] {fig/stepsim.dat};
+    \end{axis}
+  \end{tikzpicture}
+  \caption{
+    Simulated step responses along $x$ of the closed loop $\mathcal{H}_\infty$
+    controller design. In the third plot from the left, continuous lines show
+    the position error while dotted lines with the right axis are the velocity
+    in the corresponding directions.
+    \label{fig:step}
+  }
+\end{figure*}
+
+In the body frame we model the total force $\vec{F}$ acting on the UAV
+following \cite{naldi_design_2010} by considering a thrust force from the
+ducted-fan $\vec{F}_T = - k_T \omega^2 \uvec{z}$ as a function of the
+propeller's angular velocity $\omega$. Because of the geometry we approximate
+the generated air velocity field in the duct  as being constant and collinear
+to $\uvec{z}$  \cite{pereira_hover_2008}
+\begin{equation}
+  \vec{\nu} = \sqrt{\frac{F_T}{2a\varrho \pi^2}} \uvec{z}
+  = \frac{\omega}{\pi} \sqrt{\frac{k_T}{2a\varrho}} \uvec{z}.
+\end{equation}
+The \emph{drag} and \emph{lift} forces generated by the interaction of each
+flap with the air velocity field are then $\vec{F}_d = \frac{1}{2} \varrho S
+C_d\nu^2 \uvec{z}$ and $\vec{F}_\ell = \frac{1}{2} \varrho S C_\ell \nu^2
+\uvec{n}$ respectively ($\uvec{n}$ being given by the orientation of the flap).
+The density of air $\varrho$ is assumed to be constant, while the drag
+coefficients $S$, $C_d$ and $C_\ell$ depend on the angle of attack $\alpha$ of
+the flap making them quite difficult to determine. Hereinafter, under the
+assumption that $\alpha$ is small they will be approximated with $C_d = c_d
+\alpha^2 + c_{0}$, $C_\ell = c_\ell \alpha$ \cite{stengel_flight_2022,
+pereira_hover_2008} and $S$ is considered a constant. Finally, gravity adds a
+term $\vec{F}_g = mg \mt{\mx{R}} \uvec{k}$.
+
+For the total torque it is derived from the geometry that each flap induces
+$\vec{\tau}_{f} = (\frac{1}{3}a \uvec{t} + d\uvec{z}) \crossp (\vec{F}_d +
+\vec{F}_\ell)$, with $\uvec{t} = \uvec{n} \crossp \uvec{z}$. In addition there
+is also a torque induced by the gyroscopic procession $\vec{\tau}_g =
+\mx{R}^\mathsf{T} \omega J_r \uvec{k} \crossp \vec{\Omega}$ with $J_r$ being
+the inertia of the propeller with respect to its spin axis
+\cite{naldi_design_2010}. The resulting total quantities are then
+\begin{subequations}
+  \begin{align}
+    \vec{F} &= mg \mt{\mx{R}} \uvec{k}
+      -k_T\omega^2 \uvec{z} \nonumber \\
+      &\quad + \frac{\varrho S \nu^2}{2} \sum_i 
+        \left(c_d \alpha^2_i + c_0 \right) \uvec{z}
+        + c_\ell \alpha_i \uvec{n}_i, \\
+    \vec{\tau} &= \omega J_r \mx{R}^\mathsf{T} (\uvec{k} \crossp \vec{\Omega})
+      + \frac{\varrho c_\ell S \nu^2 d}{2} 
+      \uvec{z} \crossp \sum_i \alpha_i \uvec{n}_i.
+  \end{align}
+\end{subequations}
+
+% \subsection{Wind Gusts Model}
+% 
+% \texttt{TODO: add disturbance from wind modelled as simple, slowly changing
+% stochastic model following empirical approach as done in
+% \cite{chirarattananon_dynamics_2017}. Use 3D state with azimuth $\gamma$,
+% zenith $\beta$ and magnitude $W$ to generate force and torque. Maybe
+% make cyclostationary?}
+
+
+\subsection{Linearized Dynamics and Plant Model}
+
+To synthesize a controller we simplify the dynamics to a state-space LTI model
+$\dot{\vec{x}} = \mx{A}_x \vec{x} + \mx{B}_{xu} \vec{u}$, $\vec{y} =
+\mx{C}_{yx} \vec{x} + \mx{D}_{yu}\vec{u}$ with state and inputs given by
+\begin{equation}
+  \vec{x} = \mt{\begin{bmatrix}
+    \mt{\vec{P}} &
+    \mt{\dot{\vec{P}}} &
+    \mt{\vec{\Theta}} & 
+    \mt{\vec{\Omega}}
+  \end{bmatrix}},
+  \quad
+  \vec{u} = \mt{\begin{bmatrix}
+    \mt{\vec{\alpha}} & \omega
+  \end{bmatrix}}
+\end{equation}
+respectively. The linearization is performed in a stationary hovering state at
+height $h$ above the ground and yaw angle of \ang{45} or $\vec{P}_0 =
+-h\uvec{k}$, $\vec{\Theta}_0 = \frac{\pi}{4} \uvec{\psi}$, thus $x_0 =
+\mt{[\mt{\vec{P}}_0 ~ \mt{\vec{0}} ~ \mt{\vec{\Theta}}_0 ~ \mt{\vec{0}}]}$ and
+$u_0 = \mt{[ \mt{\vec{0}} ~ \omega_0 ]}$, where $\omega_0 \approx \sqrt{mg /
+k_T}$ is the propeller's angular velocity to make the UAV hover.
+
+% \subsection{Plant Model}
+
+The state is assumed to be known, as there is an inertial measurement unit with
+a dedicated sensor fusion chip onboard \cite{rothlin_optimization_2024}. Thus,
+we only need to model a measurement delay $T_m$ and the actuators. The
+linearized dynamics are extended with 2\textsuperscript{nd} order Padé
+approximant $G_{m}(s) \approx e^{-sT_m}$ for the output delay and two transfer
+functions $G_\alpha(s)$, $G_\omega (s)$ for the flaps and thruster
+respectively. Specifically, $G_\alpha(s)$ is a 2\textsuperscript{nd} order low
+pass filter (LPF) with a slight overshoot, while $G_\omega(s)$ is a first order
+LPF. All inputs and outputs are then normalized and a source of additive
+Gaussian noise $\vec{w}_\alpha \sim \mathcal{N}(\vec{0}, \mx{I})$ is introduced
+to model the wiggling in the servo motor of each flap. The resulting
+plant has 4 unobservable and 4 uncontrollable modes, however it remains both
+detectable and stabilizable.
+
+\section{Controller Design}
+\label{sec:control}
+
+% \subsection{Control Problem}
+
+The control objective is to track a piecewise constant position reference
+$\vec{r}(t) \in \mathbb{R}^3$ in the inertial frame. The controller should move
+the UAV to $\vec{r}(t)$ with control actions that do not exceed the actuator
+limits $\alpha \in (\underline{\alpha}, \overline{\alpha})$, $\omega \in [0,
+\overline{\omega})$, do not bring the angular components of the state too far
+from their set point, i.e. $\vec{\Theta} \in (-\varphi,\varphi)^3$, and do not
+exceed the velocity limits $\|\dot{\vec{P}}\|_1 \leq \overline{v}$.
+
+% \subsection{$\mathcal{H}_\infty$ Synthesis}
+
+To perform $\mathcal{H}_\infty$ synthesis the system is first rewritten in the
+form presented in \cite{skogestad_multivariable_2010}, in terms of control
+inputs $\vec{u} \in \mathbb{R}^5$, exogenous inputs $\vec{w} =
+\mt{[\mt{\vec{w}}_\alpha ~ \mt{\vec{r}}]} \in \mathbb{R}^7$, control outputs
+$\vec{y}_m = \mt{[\mt{(\vec{r} - \vec{y})} ~ \mt{\dot{\vec{P}}} ~
+\mt{\vec{\Theta}} ~ \mt{\vec{\Omega}}]}$ and errors $\vec{e} =
+\mt{[\mt{\vec{\alpha}}~\omega~\mt{(\vec{r} - \vec{y})} ~ \mt{\dot{\vec{P}}} ~
+\mt{\vec{\Theta}}]}$ and partitioned accordingly:
+\begin{equation}
+  \begin{bmatrix} \vec{e} \\ \vec{y}_m \end{bmatrix}
+    = \begin{bmatrix}
+      \mx{A}_{ew} & \mx{B}_{eu} \\
+      \mx{C}_{yw} & \mx{D}_{yu}
+    \end{bmatrix}
+  \begin{bmatrix} \vec{w} \\ \vec{u} \end{bmatrix}
+    = \mx{G}_\text{nom}
+  \begin{bmatrix} \vec{w} \\ \vec{u} \end{bmatrix}.
+\end{equation}
+
+Then for each component of the error a performance weight function is defined
+resulting in 5 transfer functions $W_{P,\alpha}(s)$, $W_{P,\omega}(s)$,
+$W_{P,\vec{P}}(s)$, $W_{P,\dot{\vec{P}}}(s)$, $W_{P,\vec{\Theta}}(s)$. The
+$\mathcal{H}_\infty$ synthesis is performed using the Riccati method yielding a
+stable controller $\mx{K}_\infty$ with $\gamma =
+\|\mathcal{L}(\mx{G}_\text{nom}, \mx{K}_\infty)\|_\infty \approx 0.9091$. Here
+we used the notation $\mathcal{L}$ for the linear fractional transformation
+interconnection.
+
+\section{Preliminary Results} \label{sec:simulation}
+
+The resulting closed loop plant $\mx{G}_\text{cl} =
+\mathcal{L}(\mx{G}_\text{nom}, \mx{K}_\infty)$ is simulated with a step of
+\qty{50}{\cm} along the $x$ spatial coordinate and shown in
+Fig.~\ref{fig:step}. Currently, it takes the UAV about 30 seconds to track a
+half meter step, while respecting the constraints on the actuators. Although
+stable and with good noise rejection characteristics the current design does
+not take into account large inaccuracies caused by the highly non linear
+equations of motion. For higher speeds necessitate higher pitch and roll
+angles, a more performant controller will need to cope with higher modelling
+errors. Hence the next controller will be designed with $\mu$-synthesis which
+can take into account the aforementioned modelling uncertainties.
+
+\printbibliography
+
+\iffalse
+\clearpage
+\appendix
+
+\subsection{Linearized Model Details}
+
+\begin{equation*}
+  A = \begin{bmatrix}
+    0 & I & 0 & 0 \\
+    0
+      & \frac{1}{m} \mt{\mx{R}} \frac{\partial \vec{F}}{\partial \dot{\vec{P}}}
+      & \frac{1}{m} \frac{\partial (\mt{\mx{R}} \vec{F})}{\partial \vec{\Theta}}
+      & \frac{1}{m} \mt{\mx{R}} \frac{\partial \vec{F}}{\partial \vec{\Omega}} \\
+    0 & 0
+      & \frac{\partial \mx{U}}{\partial \vec{\Theta}} \vec{\Omega}
+      & \mx{U} \\
+    0
+      & \minv{\mx{J}} \frac{\partial \vec{\tau}}{\partial \dot{\vec{P}}}
+      & \minv{\mx{J}} \frac{\partial \vec{\tau}}{\partial \vec{\Theta}}
+      & \minv{\mx{J}} \left[
+        \frac{\partial \vec{\tau}}{\partial \vec{\Omega}}
+        - \frac{\partial (\vec{\Omega} \crossp \mx{J} \vec{\Omega})}%
+        {\partial \vec{\Omega}} \right]
+  \end{bmatrix}
+\end{equation*}
+\fi
+
+\end{document}
+% vim:ts=2 sw=2 et spell: