diff --git a/jump_lin_id/id_paper.tex b/jump_lin_id/id_paper.tex
index ae89c0a528a9985a49357077d21fb8bfbac17574..4c99f223c8a24d5a9e94341be0ba3004a19560ae 100644
--- a/jump_lin_id/id_paper.tex
+++ b/jump_lin_id/id_paper.tex
@@ -517,7 +517,7 @@ The state of the robot arm consists of two joint coordinates, $q$, and their tim
     \setlength{\figurewidth}{0.495\linewidth}
     \setlength{\figureheight }{4cm}
     \input{figs/robot_val.tex}
-    \caption{Simulation of non-smooth robot dynamics with stiff contact -- validation data vs. sample time index. The dashed lines indicate the event times for the training data, highlighting that the model is able to deal effortless with the non-smooth friction, but inaccurately predicts the time evolution around the contact event which now occurs at a slightly different time instance.}
+    \caption{Simulation of non-smooth robot dynamics with stiff contact -- validation data vs. sample time index. The dashed lines indicate the event times for the training data, highlighting that the model is able to deal effortlessly with the non-smooth friction, but inaccurately predicts the time evolution around the contact event which now occurs at a slightly different time instance.}
     \label{fig:robot_val}
 \end{figure*}
 
diff --git a/jump_lin_id/pres/beamerthemeRegler2.sty b/jump_lin_id/pres/beamerthemeRegler2.sty
new file mode 100644
index 0000000000000000000000000000000000000000..2147726293e58365f557151430d738ebf183768a
--- /dev/null
+++ b/jump_lin_id/pres/beamerthemeRegler2.sty
@@ -0,0 +1,249 @@
+\DeclareOption{lionbackground}{\def\@beamer@option{%
+\AtBeginDocument {%
+  \pgfdeclareimage[width=70mm]{lionwhite}{LionSealWhite}
+}%
+
+\usebackgroundtemplate{{%
+  \color{palegray}\vrule height\paperheight width\paperwidth
+  \kern -\paperwidth
+  \vbox to \paperheight{%
+    \vss\kern2mm\hbox to \paperwidth{\hss\pgfuseimage{lionwhite}\hss}\vss}%
+  }%
+}
+
+\useframetitletemplate{\par\kern-1mm
+    \vbox to 10mm{\leavevmode\kern-\beamer@leftmargin
+      \colorbox{header}{\hbox to \paperwidth{\hss
+         \color{white}
+          \Large\bfseries\vrule height 7mm
+      depth 3mm width0mm\insertframetitle\strut\hss}}\kern -30mm\par\vss}%
+}%
+
+\useinnertheme[shadow=true]{rounded}
+\setbeamercolor{block title}{use=structure,fg=white,bg=structure.fg}
+
+\usefoottemplate{%
+  \vbox{\tiny%
+  \hbox{%
+  \setbox\beamer@linebox=\hbox to\paperwidth{%
+    \hbox to.5\paperwidth{\tiny\color{white}\frame@numbers\hfill\textbf{\insertshortauthor}\hskip.3cm}%
+    \hbox to.5\paperwidth{\hskip.3cm\tiny\color{white}\textbf{\insertshorttitle}\hfill}\hfill}%
+  \ht\beamer@linebox=2.625ex%
+  \dp\beamer@linebox=0pt%
+  \setbox\beamer@linebox=\vbox{\box\beamer@linebox\vskip1.125ex}%
+  \color{header}\hskip-\Gm@lmargin\vrule width.5\paperwidth
+  height\ht\beamer@linebox\color{structure}\vrule width.5\paperwidth
+  height\ht\beamer@linebox\hskip-\paperwidth%
+  \hbox{\box\beamer@linebox\hfill}\hfill\hskip-\Gm@rmargin}}}
+
+\setbeamercovered{transparent}
+
+}} % end \DeclareOption{lionbackground}
+
+\DeclareOption{liontopcorner}{\def\@beamer@option{%
+\pgfdeclareimage[width=14mm]{lionsealwhitesmall}{LionSealWhite}
+\useframetitletemplate{\par\kern-1mm
+  \vbox to 10mm{\leavevmode\kern-\beamer@leftmargin
+       \colorbox{header}{\hbox to \paperwidth{\hss
+         \color{white}
+         \Large\bfseries\vrule height 7mm  depth 3mm width0mm\relax
+         \insertframetitle\strut\hss}}\kern -30mm\par\vss}%
+     \vbox to 0pt{\kern-18mm\hbox to 0pt{\kern-12mm\relax
+              \pgfuseimage{lionsealwhitesmall}\hss}\vss}
+}%
+\useinnertheme[shadow=true]{rounded}
+\setbeamercolor{block title}{use=structure,fg=white,bg=structure.fg}
+
+\usefoottemplate{%
+  \vbox{\tiny%
+  \hbox{%
+  \setbox\beamer@linebox=\hbox to\paperwidth{%
+    \hbox to.5\paperwidth{\tiny\color{white}\frame@numbers
+          \hfill\textbf{\insertshortauthor}\hskip.3cm}%
+    \hbox to.5\paperwidth{\hskip.3cm\tiny\color{white}%
+          \textbf{\insertshorttitle}\hfill}\hfill}%
+  \ht\beamer@linebox=2.625ex%
+  \dp\beamer@linebox=0pt%
+  \setbox\beamer@linebox=\vbox{\box\beamer@linebox\vskip1.125ex}%
+  \color{header}\hskip-\Gm@lmargin\vrule width.5\paperwidth
+  height\ht\beamer@linebox\color{structure}\vrule width.5\paperwidth
+  height\ht\beamer@linebox\hskip-\paperwidth%
+  \hbox{\box\beamer@linebox\hfill}\hfill\hskip-\Gm@rmargin}}}
+
+\setbeamercovered{transparent}
+
+}} % end \DeclareOption{liontopcorner}
+
+\DeclareOption{lionheader}{\def\@beamer@option{%
+
+  \useinnertheme[shadow=true]{rounded}
+  \setbeamercolor{block title}{use=structure,fg=white,bg=structure.fg}
+  \pgfdeclareimage[width=14mm]{lionsealbronzewhite}{LionSealBronzeWhite}
+  \useframetitletemplate{%
+    \vskip 2mm
+    {\leftskip15mm%
+    \rightskip-\beamer@rightmargin plus1fill\relax
+    \advance\rightskip by0.3cm\leavevmode
+    \color{black}\Large\bfseries\insertframetitle\par \small\insertframesubtitle\vspace{-3mm}\par}
+  \vbox to 2mm{\leavevmode\kern-\beamer@leftmargin \fboxsep=0.7mm\colorbox{header}{\hbox to \paperwidth{\hss}}\vss}
+  \moveleft 3mm \vbox to 0pt{\kern-20mm\pgfuseimage{lionsealbronzewhite}\vss}
+  \vskip -7mm
+}
+
+\usefoottemplate{%
+  \vbox to 3mm{\hbox to \textwidth{\tiny
+      \hbox to 0pt{\kern -10mm\frame@numbers\hss}
+        \insertshortauthor: \insertshorttitle\hfill}\vss}}
+
+\setbeamercovered{transparent}
+}} %end \DeclareOption{lionheader}
+
+
+\DeclareOption{lionheaderLCCC}{\def\@beamer@option{%
+
+  \useinnertheme[shadow=true]{rounded}
+  \setbeamercolor{block title}{use=structure,fg=white,bg=structure.fg}
+  \pgfdeclareimage[width=14mm]{lionsealbronzewhite}{LionSealBronzeWhite}
+  \pgfdeclareimage[width=9mm]{logo-lccc}{logo-lccc}
+  \useframetitletemplate{%
+    \vskip 2mm
+    {\leftskip15mm%
+    \rightskip-\beamer@rightmargin plus1fill\relax
+    \advance\rightskip by0.3cm\leavevmode
+    \color{black}\Large\bfseries\insertframetitle\par}
+  \vbox to 2mm{\leavevmode\kern-\beamer@leftmargin \fboxsep=0.7mm\colorbox{header}{\hbox to \paperwidth{\hss}}\vss}
+  \moveleft 5mm \vbox to 0pt{\kern-20mm\pgfuseimage{lionsealbronzewhite}\vss}
+  \moveright 105mm\vbox to 0pt{\kern-25mm\pgfuseimage{logo-lccc}\vss}
+  \vskip -7mm
+}
+
+\usefoottemplate{%
+  \vbox to 3mm{\hbox to \textwidth{\tiny
+      \hbox to 0pt{\kern -10mm\frame@numbers\hss}
+        \insertshortauthor: \insertshorttitle\hfill}\vss}}
+
+\setbeamercovered{transparent}
+}} %end \DeclareOption{lionheaderLCCC}
+
+\DeclareOption{lionheaderLCCCold}{\def\@beamer@option{%
+\newsavebox\lionheadbox
+\AtBeginDocument {%
+  \pgfdeclareimage[width=116mm]{blueline}{blueline}
+  \pgfdeclareimage[width=14mm]{lionbronzejpg}{lionbronzejpg}
+  \pgfdeclareimage[width=10mm]{logo-lccc}{logo-lccc}
+  \sbox{\lionheadbox}{\vbox to 0pt{\vss \parskip=0pt\noindent
+    \pgfuseimage{lionbronzejpg}\par
+    \kern -5.3mm
+    \hbox to \textwidth{\hss\pgfuseimage{blueline}\hss}
+    \kern -13mm
+    \hbox{\kern 96mm\pgfuseimage{logo-lccc}}
+  }%
+  }%
+}
+
+\useinnertheme[shadow=true]{rounded}
+\setbeamercolor{block title}{use=structure,fg=white,bg=structure.fg}
+
+\useframetitletemplate{%
+  \vskip 5mm
+  {\leftskip17mm%
+    \rightskip-\beamer@rightmargin plus1fill\relax
+    \advance\rightskip by0.3cm\leavevmode
+   \color{black}\Large\bfseries\insertframetitle\par
+  }\usebox{\lionheadbox}\par}
+
+\usefoottemplate{%
+  \vbox to 3mm{\hbox to \textwidth{\tiny
+      \hbox to 0pt{\kern -10mm\frame@numbers\hss}
+        \insertshortauthor: \insertshorttitle\hfill}\vss}}
+
+\setbeamercovered{transparent}
+}} %end \DeclareOption{lionheaderLCCCold}
+
+
+\DeclareOption{lioncorner}{\def\@beamer@option{%
+\newsavebox\lioncornerbox
+\AtBeginDocument {%
+  \pgfdeclareimage[width=30mm]{lionbronze}{LionSealBronze}
+  \sbox{\lioncornerbox}{\vbox to 0pt{\vskip3mm\hbox to \textwidth{%
+        \hskip2mm\pgfuseimage{lionbronze}\hss}%
+  \vss}%
+  }%
+}
+
+\useframetitletemplate{%
+\vbox to 0pt{%
+  \kern68mm\hbox to 0pt{\kern93mm\pgfuseimage{lionbronze}%
+    \hss}\vss}
+  \kern-2mm
+  {\leftskip\z@ plus1fill \rightskip\z@ plus1fill
+    \color{LUblue}\Large\bfseries\insertframetitle\par}
+  {\color{bronze}\medskip\hrule height 2pt \vspace{1mm}}}
+
+\usefoottemplate{%
+  \vbox to 3mm{\hbox to \textwidth{\tiny
+      \hbox to 0pt{\kern -10mm\frame@numbers\hss}
+        \insertshortauthor: \insertshorttitle\hfill}\vss}}
+
+}} %end \DeclareOption{lioncorner}
+
+\DeclareOption{handout}{\def\@beamer@option{%
+\useframetitletemplate{\par\kern-1mm
+    \vbox to 10mm{\leavevmode\kern-\beamer@leftmargin
+      \colorbox{white}{\hbox to \paperwidth{\hss
+         \color{header}
+          \Large\bfseries\vrule height 7mm
+          depth 2mm width0mm\insertframetitle\strut\hss}}\kern
+      -30mm\par
+      \color{structure}\hrule height 1mm\vss}%
+}%
+%
+}} %end \DeclareOption{handout}
+
+\def\@beamer@option{\PackageError{'Regler'}{No theme variant specified}{}}
+
+\DeclareOption{framenumbers}{\def\frame@numbers{%
+    \quad\insertframenumber/\inserttotalframenumber}}
+\let\frame@numbers=\relax
+
+\ProcessOptions
+\usepackage{lmodern}
+\usepackage[scaled=0.9]{helvet}
+\usepackage{amsmath}
+\newenvironment{gmatrix}{\left\lgroup\begin{matrix}}{\end{matrix}\right\rgroup}
+% \usefonttheme{professionalfonts}
+% \usefonttheme[onlymath]{serif}
+
+\definecolor{bronze}{rgb}{0.61,0.38,0.08}
+\definecolor{palegray}{rgb}{0.97,0.95,0.95}
+\definecolor{header}{rgb}{0,0,0.5}
+\definecolor{LUblue}{rgb}{0,0,0.5}
+
+\setbeamercolor{structure}{fg=bronze}
+
+\parskip=\medskipamount
+\userightsidebartemplate{0pt}{}
+
+
+\AtBeginDocument {%
+  \pgfdeclareimage[width=70mm]{liongrey}{LionSealGrey}
+}%
+
+
+\@beamer@option
+
+\usetitlepagetemplate{
+  \vbox to 0pt{\kern11.2mm\relax
+      \hbox to \hsize{\hss\pgfuseimage{liongrey}\hss}\vss}
+  \vbox to \textheight{
+    \vss
+    \begin{centering}
+      {\Large\color{LUblue}\bfseries\inserttitle\par}
+      \vspace{5mm}
+      {\normalsize\textbf\insertauthor\par}
+      \vspace{5mm}
+      {\scriptsize\insertinstitute\par}%\par\vskip1em
+    \end{centering}
+    \vss
+  }
+}
diff --git a/jump_lin_id/pres/beamerthemeliontopcorner.sty b/jump_lin_id/pres/beamerthemeliontopcorner.sty
new file mode 100644
index 0000000000000000000000000000000000000000..9759a8aa13ed00cc0b753467e02a304daffa1895
--- /dev/null
+++ b/jump_lin_id/pres/beamerthemeliontopcorner.sty
@@ -0,0 +1,5 @@
+
+\PassOptionsToPackage{\CurrentOption}{beamerthemeRegler2}
+\PassOptionsToPackage{liontopcorner}{beamerthemeRegler2}
+
+\RequirePackage{beamerthemeRegler2}
diff --git a/jump_lin_id/pres/pres_idpaper.tex b/jump_lin_id/pres/pres_idpaper.tex
index 4c519ea1a424b2977267d40acae0bdbaab613c33..a996519e72bdcd2886026864e6dc38d16ceed5b8 100644
--- a/jump_lin_id/pres/pres_idpaper.tex
+++ b/jump_lin_id/pres/pres_idpaper.tex
@@ -15,6 +15,7 @@
 \usepackage{siunitx}
 \usepackage{color}
 \usepackage{pgfplots}
+\usepackage{booktabs}\usepackage{multirow}
 \usepgfplotslibrary{groupplots}
 \pgfplotsset{compat=newest}
 \usepackage{tikz}
@@ -45,10 +46,10 @@ label=center:{{$\sum$}}, minimum width=2em}}
 \setbeamercolor{item}{fg=actualbronze} % Change color of item bullet
 \setbeamercolor{block title}{use=structure,fg=white,bg=structure.fg}
 
-\title[Neural-Networks for Dynamical System Modeling]{Tangent-Space Regularization for Dynamical System  Modeling using Neural Networks}
+\title[Identification of LTV Models]{Identification of LTV Dynamical Models with\\ Smooth or Discontinuous Time Evolution \\by means of Convex Optimization}
 
 \date{\today}
-% \author[Fredrik Bagge Carlson]{\textbf{\large Fredrik Bagge Carlson}, \textnormal{Anders Robertsson, Rolf Johansson}}
+\author[Fredrik Bagge Carlson]{\textbf{\large Fredrik Bagge Carlson}, \textnormal{Anders Robertsson, Rolf Johansson}}
 \institute{Lund University, Department of Automatic Control}
 
 
@@ -68,19 +69,29 @@ label=center:{{$\sum$}}, minimum width=2em}}
 \newcommand{\cmt}[1]{{\color{yellow}{\textbf{Comment:} #1}}}
 \newcommand{\T}{^{\hspace{-0.1mm}\scriptscriptstyle \mathsf{T}}\hspace{-0.2mm}}
 \newcommand{\iT}{^{-T}\hspace{-0.6mm}}
-\newcommand{\norm}[1]{\begin{Vmatrix}#1\end{Vmatrix}_2}
+\newcommand{\normt}[1]{\begin{Vmatrix}#1\end{Vmatrix}_2}
+\newcommand{\norm}[1]{\begin{Vmatrix}#1\end{Vmatrix}}
 \newcommand{\inspace}[1]{\in \mathbb{R}^{#1}}
 \newcommand{\incspace}[1]{\in \mathbb{C}^{#1}}
 \newcommand{\card}[1]{\text{card}(#1)}
 \renewcommand{\v}{v}
 \renewcommand{\a}{\dot{v}}
 \newcommand{\amp}{A}
-\newcommand{\A}{\mathbf{A}}
-\newcommand{\w}{k}
+\newcommand{\A}{\Phi}
+\newcommand{\y}{y}
 \newcommand{\PI}{\left(\A \hspace{-0.2mm}\T\hspace{-0.1mm}\A\right)^{\hspace{-0.4mm}-1} \hspace{-1mm} \A\hspace{-0.3mm}\T}
 \newcommand{\tA}{\tilde{\mathbf{A}}}
+%\newcommand{\A}{\}
+\newcommand{\w}{k}
+\newcommand{\N}{\mathcal{N}}
 \DeclareMathOperator{\sign}{sign}
 \DeclareMathOperator*{\argmin}{arg\,min}
+\newcommand{\minimize}[1]{\underset{#1}{\text{minimize} }}
+\newcommand{\subjto}{\text{subject to }}
+\renewcommand{\vec}[1]{\operatorname{vec}{(#1)}}
+\newcommand{\diag}[1]{\operatorname{diag}{(#1)}}
+\newcommand{\bmatrixx}[1]{\begin{bmatrix}#1\end{bmatrix}}
+
 
 
 \begin{document}
@@ -94,31 +105,381 @@ label=center:{{$\sum$}}, minimum width=2em}}
 
 %====================================================================
 %====================================================================
-\begin{frame}{Introduction}
-    Dynamical control systems are often described by differential state-equations
-    $$\dot x(t) = f_c(x(t), u(t))$$
-    where $x$ is the state, $u$ is the input
-    \begin{block}{Example -- Robot}
-        $$\ddot x = M^{-1}(x)  \big( C(x,\dot x)\dot x + G(x) + F(\dot x) - u \big)$$
-    \end{block}
+\begin{frame}{LTI identification}
+    We start by considering the case of identification of the parameters in an LTI model on the form
+    \begin{equation}
+        x_{t+1} = A x_t + B u_t + v_t, \quad t \in [1,T]
+    \end{equation}
+    where $x\inspace{n}$ and $u\inspace{m}$ are the state and input respectively.
+\end{frame}
+
+
+\begin{frame}{}{}
+    $\y = \A\w$, and arrange the data according to
+    \begin{align*}
+        \y &=
+        \begin{bmatrix}
+            {x_1} \\ \vdots \\ {x_T}
+        \end{bmatrix}
+        & &\inspace{Tn} \\
+        \w &= \vec{\bmatrixx{A\T & B\T}} & &\inspace{K}\\[0.2em]
+        \A &=
+        \begin{bmatrix}
+            I_n \otimes x_0\T & I_n \otimes u_0\T \\
+            \vdots & \vdots\\
+            I_n \otimes x_{T-1}\T & I_n \otimes u_{T-1}\T
+        \end{bmatrix}
+        & &\in \mathbb{R}^{Tn\times K}
+    \end{align*}
+
+    \begin{align}
+        \w^* &= \argmin_{\w} \normt{\A \w - \y}^2 \label{eq:lscost}\\
+        ~ &= \PI \y \label{eq:ls}
+    \end{align}
+
+\end{frame}
+
+
+
+
+%====================================================================
+%====================================================================
+\begin{frame}{Time-varying dynamics}
+    We now extend our view to systems where the dynamics change with time. We limit the scope of this article to models on the form
+    \begin{equation}
+        \label{eq:tvk}
+        \begin{split}
+            x_{t+1} &= A_t x_t + B_t u_t + v_t\\
+            \w_t &= \vec{\bmatrixx{A_t\T & B_t\T}}
+        \end{split}
+    \end{equation}
     \pause
+    where the parameters $\w$ are assumed to evolve according to the dynamical system
+    \begin{equation}
+        \label{eq:dynsys}
+        \begin{split}
+            k_{t+1} &= H_t k_t + w_t\\
+            y_t &= \big(I_n \otimes \bmatrixx{x_t\T & u_t\T}\big) \w_t
+        \end{split}
+    \end{equation}
+\end{frame}
 
-    Discretization (sampling) leads to
-    $$x_{t+1} = f(x_t, u_t)$$
 
-    \begin{block}{Objective 1}
-        Learn the function $f$
-        $$x_{t+1} = f(x_t, u_t)$$
-    \end{block}
 
+
+%====================================================================
+%====================================================================
+\begin{frame}{Trend filtering}
+    An important class of identification methods that has been popularized lately is \emph{trend filtering} methods~\footfullcite{kim2009ell_1, tibshirani2014adaptive}.
+
+    As a simple example, consider the reconstruction $\hat y$ of a noisy signal $y = \{y_t\inspace{}\}_{t=1}^T$ with piecewise constant segments.
+    \begin{equation*} \label{eq:tf}
+        \minimize{\hat{y}} \normt{y-\hat{y}}^2 + \lambda\sum_t |\hat{y}_{t+1} - \hat{y}_t|
+    \end{equation*}
+    \begin{itemize}
+        \item Fitness function
+        \item (Sparsity promoting) Regularization
+        \item Convex
+    \end{itemize}
 \end{frame}
 
 
+%====================================================================
+%====================================================================
+\begin{frame}{Regularization term intuition}{}
+    figure
+
+    The 1-norm is a \emph{sparsity-promoting} penalty, hence a solution in which only a small number of non-zero first-order time differences in the model parameters is favored, i.e., a piecewise constant dynamics evolution.
+\end{frame}
+
+
+
+%====================================================================
+%====================================================================
+\begin{frame}{Low-frequency time evolution}
+    A slowly varying signal is characterized by \emph{small first-order time differences}.
+    \pause
+
+    \begin{equation} \label{eq:slow}
+        \minimize{\w} \normt{\y-\hat{\y}}^2 + \lambda^2\sum_t \normt{\w_{t+1} - \w_{t}}^2
+    \end{equation}
+    \pause
+
+    \begin{align}\label{eq:closedform}
+        \tilde{\w}^* &= (\tilde{\A}\T\tilde{\A} + \lambda^2 D_1\T D_1)^{-1}\tilde{\A}\T \tilde{Y}\\
+        \tilde{\w} &= \operatorname{vec}(\w_1, ...\,, \w_T)\nonumber
+    \end{align}
+\end{frame}
+
+
+
+
+%====================================================================
+%====================================================================
+\begin{frame}{Smooth time evolution}
+    A smoothly varying signal is characterized by \emph{small second-order time differences}.
+    \pause
+
+    \begin{equation} \label{eq:smooth}
+        \minimize{\w} \normt{\y-\hat{\y}}^2 + \lambda^2\sum_t \normt{\w_{t+2} -2 \w_{t+1} + \w_t}^2
+    \end{equation}
+\end{frame}
+
+
+%====================================================================
+%====================================================================
+\begin{frame}{Piecewise constant time evolution}\label{sec:pwconstant}
+    A signal which is mostly flat, with a small number of distinct level changes, is characterized by a \emph{sparse first-order time difference}.
+    \pause
+
+
+    \begin{equation} \label{eq:pwconstant}
+        \minimize{\w} \normt{\y-\hat{\y}}^2 + \lambda\sum_t \normt{ \w_{t+1} - \w_t}
+    \end{equation}
+    \pause
+
+    We can give \labelcref{eq:pwconstant} an interpretation as a \emph{grouped-lasso} cost function.
+
+    Penalty on the 1-norm on the \emph{length} of the difference vectors $\w_{t+1} - \w_t$ since $\norm{\normt{\cdot}}_1 = \normt{\cdot}$.
+    \pause
+
+\end{frame}
+
+\begin{frame}{}{}
+
+    At a first glance, one might consider the formulation
+    \begin{equation} \label{eq:pwconstant_naive}
+        \minimize{\w} \normt{\y-\hat{\y}}^2 + \lambda\sum_t \norm{\w_{t+1} - \w_t}_1
+    \end{equation}
+    \pause
+
+    changes to different entries of $\w_t$ would not occur at the same time instants.
+\end{frame}
+
+
+%====================================================================
+%====================================================================
+\begin{frame}{Implementation}
+    Due to the non-squared norm penalty $\sum_t \normt{ \w_{t+1} - \w_t}$, problem \labelcref{eq:pwconstant} is significantly harder to solve than \labelcref{eq:smooth}.
+
+    An efficient implementation using the linearized ADMM algorithm \footfullcite{parikh2014proximal} is made available in the accompanying repository.
+
+    \url{https://github.com/baggepinnen/LTVModels.jl}
+\end{frame}
+
+
+
+%====================================================================
+%====================================================================
+\begin{frame}{Summary}
+    The proposed optimization problems are summarized in~\cref{tab:opts}.
+
+    \begin{table}[]
+        \centering
+        \caption{Summary of optimization problem formulations. $D_n$ refers to parameter vector time-differentiation of order $n$.}
+        \label{tab:opts}
+        \begin{tabular}{@{}lll@{}}
+            \toprule
+            Norm & $D_n$ & Result    \\ \midrule
+            1    & 1         & Small number of steps (piecewise constant)     \\
+            1    & 2         & Small number of bends (piecewise affine)    \\
+            2    & 1         & Small steps (slowly varying) \\
+            2    & 2         & Small bends (smooth)  \\ \bottomrule
+        \end{tabular}
+    \end{table}
+
+\end{frame}
+
+
+
+
+
+%====================================================================
+%====================================================================
+\begin{frame}{Example -- Jump-linear system}
+    We now consider a simulated example. Change in dynamics, from
+    $$A_t = \left[
+    \begin{array}{cc}
+        0.95 & 0.1 \\
+        0.0 & 0.95 \\
+    \end{array}
+    \right], \quad B_t = \left[
+    \begin{array}{c}
+        0.2 \\
+        1.0 \\
+    \end{array}
+    \right]
+    $$
+    to $$A_t = \left[
+    \begin{array}{cc}
+        0.5 & 0.05 \\
+        0.0 & 0.5 \\
+    \end{array}
+    \right], \quad B_t = \left[
+    \begin{array}{c}
+        0.2 \\
+        1.0 \\
+    \end{array}
+    \right]
+    $$
+    occurred at $t=200$.
+
+    \begin{description}
+        \item[Input] $u \sim \N(0, 1)$
+        \item[state transition noise and measurement noise] $\N(0, 0.2^2)$
+    \end{description}
+
+\end{frame}
+
+\begin{frame}{Example -- Jump-linear system}
+    \begin{figure}
+        \centering
+        \setlength{\figurewidth}{0.99\linewidth}
+        \setlength{\figureheight }{5.5cm}
+        \input{../figs/ss.tex}
+        \caption{True values are shown with dashed, black lines. Gaussian state-transition and measurement noise with $\sigma = 0.2$ were added.}
+        \label{fig:ss}
+    \end{figure}
+
+\end{frame}
+
+
+
+%====================================================================
+%====================================================================
+\begin{frame}{Example -- Robot arm}
+    \begin{itemize}
+        \item Non-smooth dynamics
+        \item Discontinuous Coulomb friction
+        \item Stiff contact with environment
+    \end{itemize}
+
+    % The state of the robot arm consists of two joint coordinates, $q$, and their time derivatives, $\dot q$. \Cref{fig:robot_train} illustrates the state trajectories, control torques and simulations of a model estimated by solving~\labelcref{eq:pwconstant}. The figure clearly illustrates that the model is able to capture the dynamics both during the non-smooth sign change of the velocity, but also during the establishment of the stiff contact. The learned dynamics of the contact is however time-dependent, which is, in some situations, a drawback of the model and is illustrated in \Cref{fig:robot_val}, where the model is used on a validation trajectory where a different noise sequence was added to the control torque. Due to the novel input signal, the contact is established at a different time-instant and as a consequence, there is an error transient in the simulated data.
+
+\end{frame}
+%====================================================================
+%====================================================================
+\begin{frame}{Robot -- Training trajectory}{}
+
+    \begin{figure}
+        \centering
+        \pgfplotsset{every axis/.append style={
+        label style={font=\tiny},
+        legend style={font=\tiny, draw=none},
+        tick label style={font=\tiny}
+        }}
+        \setlength{\figurewidth}{0.495\linewidth}
+        \setlength{\figureheight }{2.7cm}
+        \input{../figs/robot_train.tex}
+        \label{fig:robot_train}
+    \end{figure}
+\end{frame}
+%====================================================================
+%====================================================================
+\begin{frame}{Robot -- Validation trajectory}{}
+    \begin{figure}
+        \centering
+        \pgfplotsset{every axis/.append style={
+        label style={font=\tiny},
+        legend style={font=\tiny, draw=none},
+        tick label style={font=\tiny}
+        }}
+        \setlength{\figurewidth}{0.495\linewidth}
+        \setlength{\figureheight }{3cm}
+        \input{../figs/robot_val.tex}
+        \caption{Validation data vs. sample time index. The dashed lines indicate the event times for the training data, highlighting that the model is able to deal effortless with the non-smooth friction, but inaccurately predicts the time evolution around the contact event which now occurs at a slightly different time instance.}
+        \label{fig:robot_val}
+    \end{figure}
+\end{frame}
+
+
+
+%====================================================================
+%====================================================================
+\begin{frame}{Example -- Reinforcement learning} \label{sec:rl}
+    \begin{itemize}
+        \item Identify LTV dynamics models for reinforcement learning
+        \item Dampen oscillations of a pendulum on a cart
+        \item Quadratic cost on states and control
+        \begin{enumerate}
+            \item fit a dynamics model along the last obtained trajectory
+            \item optimize the cost function under
+            the model using iterative LQG (differential dynamic programming)\footnote{Implementation made available at
+            \href{github.com/baggepinnen/DifferentialDynamicProgramming.jl}{github.com/baggepinnen/DifferentialDynamicProgramming.jl}}
+            \item In order to stay close to the validity region of the linear model, we put bounds on the deviation between each new trajectory and the last trajectory.
+        \end{enumerate}
+    \end{itemize}
+
+\end{frame}
+%====================================================================
+%====================================================================
+\begin{frame}{Example -- Reinforcement learning}{}
+
+    We compare three different models
+    \begin{itemize}
+        \item The ground truth system model
+        \item LTV model (obtained by solving \labelcref{eq:smooth})
+        \item LTI model
+    \end{itemize}
+
+    The total cost over $T=500$ time steps is shown as a function of learning iteration.
+    \begin{figure}[htp]
+        \centering
+        \setlength{\figurewidth}{0.99\linewidth}
+        \setlength{\figureheight }{4.5cm}
+        \pgfplotsset{every axis/.append style={
+        legend style={draw=black!20!white}
+        }}
+        \input{../figs/ilc.tex}
+    \end{figure}
+\end{frame}
+
+
+%====================================================================
+%====================================================================
+\begin{frame}{Discussion}{}
+    \begin{itemize}
+        \item[\nice{+}] The methods presented extend directly to nonlinear models that remain \emph{linear in the parameters}.
+        \item[\bad{-}] A first-order approximation to a nonlinear system is not guaranteed to generalize well as deviations from the trajectory become large.
+        \item All assumptions over time
+    \end{itemize}
+
+\end{frame}
+
+
+%====================================================================
+%====================================================================
+\begin{frame}{Discussion -- Reinforcement learning}{}
+    \begin{itemize}
+        \item For iterative learning control and trajectory centric reinforcement learning, a first-order approximation to the dynamics is used for efficient optimization
+        \item Validity of the approximation is ensured by incorporating penalties or constraints between two consecutive trajectories.
+        \item[\nice{+}] This makes the proposed identification methods attractive in applications such as guided policy search (GPS)~\footfullcite{levine2013guided, levine2015learning} and non-linear iterative learning control (ILC)~\footfullcite{bristow2006survey}, where they can lead to dramatically decreased sample complexity.
+    \end{itemize}
+\end{frame}
+
+
+
+%====================================================================
+%====================================================================
+\begin{frame}{Conclusions}{}
+    \begin{itemize}
+        \item Framework for identification of linear, time-varying models along trajectories of nonlinear dynamical systems using convex optimization
+        \item Applications within trajectory-centric, model-based reinforcement learning, iterative learning control (ILC), and jump-linear system identification
+    \end{itemize}
+    \pause
+
+    In the paper
+    \begin{itemize}
+        \item Analysis of identifyability
+        \item Kalman smoother for efficient identification
+    \end{itemize}
+\end{frame}
+
 
 %====================================================================
 %====================================================================
 \begin{frame}{Open source}{}
-    Code to train the models presented in this talk available at
+    Code to train the models and reproduce examples presented in this talk available at
     \url{https://github.com/baggepinnen/LTVModels.jl}
 \end{frame}