From 7e92f609ea99d9c503135f0a75760bf7cc39b69f Mon Sep 17 00:00:00 2001
From: baggepinnen <cont-frb@ulund.org>
Date: Thu, 14 Jun 2018 18:40:40 +0200
Subject: [PATCH] add pres figs

---
 jump_lin_id/figs/force.tex                    | 2298 ---------------
 jump_lin_id/figs/states.tex                   | 2454 -----------------
 jump_lin_id/figs/surf.png                     |  Bin 150604 -> 0 bytes
 jump_lin_id/pres/figs/kalman-0.png            |  Bin 0 -> 410155 bytes
 jump_lin_id/pres/figs/kalman-1.png            |  Bin 0 -> 392182 bytes
 jump_lin_id/pres/figs/kalman-2.png            |  Bin 0 -> 378465 bytes
 jump_lin_id/pres/figs/kalman-3.png            |  Bin 0 -> 368065 bytes
 jump_lin_id/pres/figs/kalman-4.png            |  Bin 0 -> 354905 bytes
 jump_lin_id/pres/figs/kalman-5.png            |  Bin 0 -> 340380 bytes
 jump_lin_id/pres/figs/kalman-6.png            |  Bin 0 -> 329883 bytes
 jump_lin_id/pres/figs/kalman-7.png            |  Bin 0 -> 320250 bytes
 jump_lin_id/pres/figs/kalman-8.png            |  Bin 0 -> 308975 bytes
 jump_lin_id/pres/figs/kalman-9.png            |  Bin 0 -> 302078 bytes
 jump_lin_id/pres/figs/pendcart_draw.jpg       |  Bin 0 -> 49156 bytes
 .../pres/figs/pres_idpaper-figure0.dpth       |    0
 .../pres/figs/pres_idpaper-figure0.pdf        |  Bin 0 -> 24879 bytes
 .../pres/figs/pres_idpaper-figure1.dpth       |    0
 .../pres/figs/pres_idpaper-figure1.pdf        |  Bin 0 -> 22593 bytes
 .../pres/figs/pres_idpaper-figure2.dpth       |    0
 .../pres/figs/pres_idpaper-figure2.pdf        |  Bin 0 -> 22585 bytes
 .../pres/figs/pres_idpaper-figure3.dpth       |    0
 .../pres/figs/pres_idpaper-figure3.pdf        |  Bin 0 -> 40623 bytes
 .../pres/figs/pres_idpaper-figure4.dpth       |    0
 .../pres/figs/pres_idpaper-figure4.pdf        |  Bin 0 -> 33116 bytes
 jump_lin_id/pres/figs/qr.png                  |  Bin 0 -> 2243 bytes
 jump_lin_id/pres/figs/robot.tex               |   90 +
 jump_lin_id/pres/figs/robot_draw.jpg          |  Bin 0 -> 22529 bytes
 jump_lin_id/pres/figs/robot_draw.pdf          |  Bin 0 -> 11791 bytes
 jump_lin_id/pres/pres_idpaper.pdf             |  Bin 480991 -> 2162977 bytes
 jump_lin_id/pres/pres_idpaper.tex             |  359 ++-
 30 files changed, 300 insertions(+), 4901 deletions(-)
 delete mode 100644 jump_lin_id/figs/force.tex
 delete mode 100644 jump_lin_id/figs/states.tex
 delete mode 100644 jump_lin_id/figs/surf.png
 create mode 100644 jump_lin_id/pres/figs/kalman-0.png
 create mode 100644 jump_lin_id/pres/figs/kalman-1.png
 create mode 100644 jump_lin_id/pres/figs/kalman-2.png
 create mode 100644 jump_lin_id/pres/figs/kalman-3.png
 create mode 100644 jump_lin_id/pres/figs/kalman-4.png
 create mode 100644 jump_lin_id/pres/figs/kalman-5.png
 create mode 100644 jump_lin_id/pres/figs/kalman-6.png
 create mode 100644 jump_lin_id/pres/figs/kalman-7.png
 create mode 100644 jump_lin_id/pres/figs/kalman-8.png
 create mode 100644 jump_lin_id/pres/figs/kalman-9.png
 create mode 100644 jump_lin_id/pres/figs/pendcart_draw.jpg
 create mode 100644 jump_lin_id/pres/figs/pres_idpaper-figure0.dpth
 create mode 100644 jump_lin_id/pres/figs/pres_idpaper-figure0.pdf
 create mode 100644 jump_lin_id/pres/figs/pres_idpaper-figure1.dpth
 create mode 100644 jump_lin_id/pres/figs/pres_idpaper-figure1.pdf
 create mode 100644 jump_lin_id/pres/figs/pres_idpaper-figure2.dpth
 create mode 100644 jump_lin_id/pres/figs/pres_idpaper-figure2.pdf
 create mode 100644 jump_lin_id/pres/figs/pres_idpaper-figure3.dpth
 create mode 100644 jump_lin_id/pres/figs/pres_idpaper-figure3.pdf
 create mode 100644 jump_lin_id/pres/figs/pres_idpaper-figure4.dpth
 create mode 100644 jump_lin_id/pres/figs/pres_idpaper-figure4.pdf
 create mode 100644 jump_lin_id/pres/figs/qr.png
 create mode 100644 jump_lin_id/pres/figs/robot.tex
 create mode 100644 jump_lin_id/pres/figs/robot_draw.jpg
 create mode 100644 jump_lin_id/pres/figs/robot_draw.pdf

diff --git a/jump_lin_id/figs/force.tex b/jump_lin_id/figs/force.tex
deleted file mode 100644
index 97a93ac..0000000
--- a/jump_lin_id/figs/force.tex
+++ /dev/null
@@ -1,2298 +0,0 @@
-% This file was created by matplotlib2tikz v0.6.2.
-\begin{tikzpicture}
-
-\definecolor{color1}{rgb}{0.88887350027252,0.43564919034819,0.278122936141944}
-\definecolor{color0}{rgb}{0,0.605603161175225,0.978680117569607}
-\definecolor{color3}{rgb}{0.764440175493436,0.444111779468777,0.824297535923276}
-\definecolor{color2}{rgb}{0.242224297852199,0.64327509315763,0.304448651534115}
-\definecolor{color5}{rgb}{4.82118162326017e-07,0.665758981292356,0.680996951870795}
-\definecolor{color4}{rgb}{0.675543957211406,0.555662332204581,0.094234336266395}
-
-\begin{axis}[
-title={Force/torque},
-xmin=1, xmax=372,
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-axis on top,
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diff --git a/jump_lin_id/figs/states.tex b/jump_lin_id/figs/states.tex
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index 3213996..0000000
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diff --git a/jump_lin_id/pres/pres_idpaper.pdf b/jump_lin_id/pres/pres_idpaper.pdf
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diff --git a/jump_lin_id/pres/pres_idpaper.tex b/jump_lin_id/pres/pres_idpaper.tex
index dfae9ee..aadd701 100644
--- a/jump_lin_id/pres/pres_idpaper.tex
+++ b/jump_lin_id/pres/pres_idpaper.tex
@@ -118,6 +118,44 @@ label=center:{{$\sum$}}, minimum width=2em}}
 \end{frame}
 
 
+%====================================================================
+%====================================================================
+\begin{frame}{Outline}{}
+    \begin{itemize}
+        \item Introduce a number of LTV identification methods
+        \item Use of methods for reinforcement learning
+    \end{itemize}
+\end{frame}
+
+
+%====================================================================
+%====================================================================
+\begin{frame}{Motivating example}{}
+    \begin{block}{LQR}
+        \begin{itemize}
+            \item Control signal trajectory
+            \item Feedback gain
+            \item Dynamic programming
+            \item Linear systems
+            \item Quadratic cost
+        \end{itemize}
+    \end{block}
+    \pause
+
+    \begin{block}{Iterative LQR}
+        \begin{itemize}
+            \item Linearize system
+            \item Differential Dynamic programming\footfullcite{levine2013guided}
+            \item Nonlinear systems
+            \item Stepsize or constraint
+        \end{itemize}
+
+    \end{block}
+
+\end{frame}
+
+
+
 %====================================================================
 %====================================================================
 \begin{frame}{Outline}{}
@@ -150,32 +188,32 @@ label=center:{{$\sum$}}, minimum width=2em}}
 
 
 
-\begin{frame}{LTI identification}{}
-    Linear in the parameters, can be written $\y = \A\w$
-    \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*}
-    \pause
-
-    Closed form solution
-    \begin{align}
-        \w^* &= \argmin_{\w} \normt{\y - \A \w}^2 \label{eq:lscost}\\
-        ~ &= \PI \y \label{eq:ls}
-    \end{align}
-
-\end{frame}
+% \begin{frame}{LTI identification}{}
+%     Linear in the parameters, can be written $\y = \A\w$
+%     \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*}
+%     \pause
+%
+%     Closed form solution
+%     \begin{align}
+%         \w^* &= \argmin_{\w} \normt{\y - \A \w}^2 \label{eq:lscost}\\
+%         ~ &= \PI \y \label{eq:ls}
+%     \end{align}
+%
+% \end{frame}
 
 
 
@@ -195,17 +233,17 @@ label=center:{{$\sum$}}, minimum width=2em}}
     \end{block}
     \pause
 
-    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}
-    \pause
-
-    Free to choose $H$, e.g., $H = I$
+    % 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}
+    % \pause
+    %
+    % Free to choose $H$, e.g., $H = I$
 \end{frame}
 
 
@@ -233,14 +271,16 @@ label=center:{{$\sum$}}, minimum width=2em}}
 %====================================================================
 %====================================================================
 \begin{frame}{Trend filtering}
-    Class of identification methods for signal reconstruction, \emph{trend filtering}~\footfullcite{kim2009ell_1, tibshirani2014adaptive}.
+    Class of identification methods for signal reconstruction~\footfullcite{kim2009ell_1, tibshirani2014adaptive}.
     \pause
 
-    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*}
-    \pause
+    \begin{block}{Example}
+        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*}
+    \end{block}
+
     \begin{itemize}
         \item Fitness function
         \item (Sparsity promoting) Regularization
@@ -253,33 +293,35 @@ label=center:{{$\sum$}}, minimum width=2em}}
 %====================================================================
 \begin{frame}{Norms for Regularization}{}
     \begin{description}
-        \item[$\norm{k}_1$ 1-norm] \emph{Sparsity-promoting} penalty.\\ A solution with a small number of non-zero entries in $k$ is favored.\vspace{2mm}
-        \item[$\norm{k}_2$ 2-norm] Penalizes large entries in $k$. \\Does not care about small, non-zero entries.
+        \item[$\norm{k}_1$] \emph{Sparsity-promoting} penalty.\\ A solution with a small number of non-zero entries in $k$ is favored.\vspace{2mm}
+        \item[$\sum_t\norm{k_t}_2^2$] Penalizes large entries in $k_t$. \\Does not care about small, non-zero entries.
+        \item[$\sum_t\norm{k_t}_2$] Penalizes non-zero $k_t$. \\ Grouped lasso.
     \end{description}
 
 \end{frame}
 
 
 
+
+
+
 %====================================================================
 %====================================================================
-\begin{frame}{Identification methods}{}
+\begin{frame}{Low-frequency time evolution}
+    System with slowly varying dynamics
     \begin{itemize}
-        \item Introduce a number of optimization problems
-        \item The difference lies in the regularization term
-        \item The regularization term can be given statistical interpretation
+        \item Temperature change
+        \item Smooth nonlinearities
+        \item Implicit assumption: parameters $k$ evolve as Brownian motion %($H = I$)
     \end{itemize}
+    \centering
+    \includegraphics[width=0.5\linewidth]{figs/kalman-0.png}
 \end{frame}
 
-
-%====================================================================
-%====================================================================
 \begin{frame}{Low-frequency time evolution}
     A slowly varying signal is characterized by \emph{small first-order time differences}.
     \pause
 
-    Assume parameters $k$ evolve according to Brownian motion ($H = I$)
-
     \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}
@@ -292,7 +334,19 @@ label=center:{{$\sum$}}, minimum width=2em}}
     \end{align}
     \pause
 
-    Parameters found efficiently by Dynamic programming!
+    \begin{itemize}
+        \item Parameters found efficiently by Dynamic programming!
+        \item Generalization in paper.
+    \end{itemize}
+\end{frame}
+
+
+\begin{frame}{Low-frequency time evolution}
+    \centering
+    \only<1>{\includegraphics[width=0.8\linewidth]{figs/kalman-1.png}}
+    \only<2>{\includegraphics[width=0.8\linewidth]{figs/kalman-4.png}}
+    \only<3>{\includegraphics[width=0.8\linewidth]{figs/kalman-7.png}}
+    \only<4>{\includegraphics[width=0.8\linewidth]{figs/kalman-9.png}}
 \end{frame}
 
 
@@ -300,25 +354,35 @@ label=center:{{$\sum$}}, minimum width=2em}}
 
 %====================================================================
 %====================================================================
-\begin{frame}{Smooth time evolution}
-    A smoothly varying signal is characterized by \emph{small second-order time differences}.
-    \pause
+% \begin{frame}{Smooth time evolution}
+%     A smoothly varying signal is characterized by \emph{small second-order time differences}.
+%     \pause
+%
+%     Parameters $k$ are integrated twice (inertia)
+%     \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}
+%     \pause
+%
+%     \begin{itemize}
+%         \item Closed-form solution and solution (intractable)
+%         \item Dynamic programming solution (efficient)
+%     \end{itemize}
+% \end{frame}
 
-    Parameters $k$ are integrated twice (inertia)
-    \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}
-    \pause
 
+%====================================================================
+%====================================================================
+\begin{frame}{Piecewise constant time evolution}
+    Piecewise constant dynamics
     \begin{itemize}
-        \item Closed-form solution and solution (intractable)
-        \item Dynamic programming solution (efficient)
+        \item Abrupt failure
+        \item Change of operation mode
+        \item Non-holonomic constraints
+        \item Number of change points known / unknown
     \end{itemize}
 \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
@@ -331,8 +395,6 @@ label=center:{{$\sum$}}, minimum width=2em}}
 
     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}
 
@@ -351,38 +413,13 @@ label=center:{{$\sum$}}, minimum width=2em}}
 %====================================================================
 %====================================================================
 \begin{frame}{Piecewise constant time evolution -- 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}.
+    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:slow}.
 
     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}
-    \begin{table}[]
-        \centering
-        \caption{Summary of optimization problem formulations. \hspace{\textwidth} $D_n$ 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}
@@ -439,6 +476,80 @@ label=center:{{$\sum$}}, minimum width=2em}}
 
 
 
+
+% %====================================================================
+% %====================================================================
+% \begin{frame}{Summary}
+%     \begin{table}[]
+%         \centering
+%         \caption{Summary of optimization problem formulations. \hspace{\textwidth} $D_n$ 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 -- Reinforcement learning} \label{sec:rl}
+    \begin{itemize}
+        \item Identify LTV models for reinforcement learning
+        \item Dampen oscillations of a pendulum on a cart
+        \item Quadratic cost on states and control
+        \begin{enumerate}
+            \item Fit model along trajectory
+            \item Optimize the cost function under
+            model using iterative LQG (DDP)\footnote{Implementation made available at
+            \href{github.com/baggepinnen/DifferentialDynamicProgramming.jl}{github.com/baggepinnen/DifferentialDynamicProgramming.jl}}
+            \item Bounds on the deviation between optimized trajectory and previous trajectory.
+        \end{enumerate}
+    \end{itemize}
+    \centering
+    \vspace{-5mm}\hspace{5cm}\includegraphics[width=0.3\linewidth]{figs/pendcart_draw.jpg}
+\end{frame}
+%====================================================================
+%====================================================================
+\begin{frame}{Example -- Reinforcement learning}{}
+    \footnotetext{$\normt{\w_{t+1} - \w_t}^2$}
+    \small
+    \begin{columns}
+        \begin{column}{0.5\textwidth}
+            We compare three different models
+            \begin{itemize}
+                \item The ground truth system model
+                \item LTV model \labelcref{eq:slow}\footnotemark
+                \item LTI model
+            \end{itemize}
+        \end{column}
+        \pause
+        \begin{column}{0.6\textwidth}
+            \begin{figure}[htp]
+                \centering
+                \setlength{\figurewidth}{0.99\linewidth}
+                \setlength{\figureheight }{5cm}
+                \pgfplotsset{every axis/.append style={
+                legend style={draw=black!20!white, yshift=1mm},
+                title style={yshift=1.3mm},
+                }}
+                \input{../figs/ilc.tex}
+            \end{figure}
+        \end{column}
+    \end{columns}
+
+\end{frame}
+
+
+
+
 %====================================================================
 %====================================================================
 \begin{frame}{Example -- Robot arm}
@@ -506,57 +617,6 @@ label=center:{{$\sum$}}, minimum width=2em}}
 \end{frame}
 
 
-
-%====================================================================
-%====================================================================
-\begin{frame}{Example -- Reinforcement learning} \label{sec:rl}
-    \begin{itemize}
-        \item Identify LTV models for reinforcement learning
-        \item Dampen oscillations of a pendulum on a cart
-        \item Quadratic cost on states and control
-        \begin{enumerate}
-            \item Fit model along trajectory
-            \item Optimize the cost function under
-            model using iterative LQG (DDP)\footnote{Implementation made available at
-            \href{github.com/baggepinnen/DifferentialDynamicProgramming.jl}{github.com/baggepinnen/DifferentialDynamicProgramming.jl}}
-            \item Bounds on the deviation between optimized trajectory and previous trajectory.
-        \end{enumerate}
-    \end{itemize}
-    \centering
-    \vspace{-5mm}\hspace{5cm}\includegraphics[width=0.3\linewidth]{figs/pendcart_draw.jpg}
-\end{frame}
-%====================================================================
-%====================================================================
-\begin{frame}{Example -- Reinforcement learning}{}
-    \footnotetext{$\normt{\w_{t+1} - \w_t}^2$}
-    \small
-    \begin{columns}
-        \begin{column}{0.5\textwidth}
-            We compare three different models
-            \begin{itemize}
-                \item The ground truth system model
-                \item LTV model \labelcref{eq:slow}\footnotemark
-                \item LTI model
-            \end{itemize}
-        \end{column}
-        \pause
-        \begin{column}{0.6\textwidth}
-            \begin{figure}[htp]
-                \centering
-                \setlength{\figurewidth}{0.99\linewidth}
-                \setlength{\figureheight }{5cm}
-                \pgfplotsset{every axis/.append style={
-                legend style={draw=black!20!white, yshift=1mm},
-                title style={yshift=1.3mm},
-                }}
-                \input{../figs/ilc.tex}
-            \end{figure}
-        \end{column}
-    \end{columns}
-
-\end{frame}
-
-
 %====================================================================
 %====================================================================
 \begin{frame}{Discussion}{}
@@ -574,8 +634,8 @@ label=center:{{$\sum$}}, minimum width=2em}}
 \begin{frame}{Discussion -- Reinforcement learning}{}
     \begin{itemize}
         \item In ILC and trajectory centric reinforcement learning, a first-order approximation to the dynamics is used for efficient optimization
-        \item Validity of the approximation ensured by constraints between two consecutive trajectories.
-        \item[\nice{+}] Proposed identification methods attractive in applications such as guided policy search (GPS)~\footfullcite{levine2013guided, levine2015learning} and non-linear ILC (ILC)~\footfullcite{bristow2006survey}. \\
+        \item Validity of approximation ensured by constraints between two consecutive trajectories.
+        \item[\nice{+}] Useful in applications such as guided policy search (GPS)~\footfullcite{levine2013guided, levine2015learning} and non-linear ILC (ILC)~\footfullcite{bristow2006survey}. \\
         Can lead to dramatically decreased sample complexity.
     \end{itemize}
 \end{frame}
@@ -600,6 +660,7 @@ label=center:{{$\sum$}}, minimum width=2em}}
     \begin{itemize}
         \item Analysis of identifyability
         \item Kalman smoother for efficient identification
+        \item More identification methods
     \end{itemize}
 \end{frame}
 
-- 
GitLab