From bc0906b59230929e3631c10b109dc8050bf53934 Mon Sep 17 00:00:00 2001
From: baggepinnen <cont-frb@ulund.org>
Date: Mon, 14 May 2018 13:11:30 +0200
Subject: [PATCH] update slides
---
jump_lin_id/bibtexfile.bib | 22 ++
jump_lin_id/pres/beamerthemeRegler2.sty | 162 +-------------
jump_lin_id/pres/pres_idpaper.tex | 271 ++++++++++++++++++------
3 files changed, 229 insertions(+), 226 deletions(-)
diff --git a/jump_lin_id/bibtexfile.bib b/jump_lin_id/bibtexfile.bib
index b1bf7be..d74999c 100644
--- a/jump_lin_id/bibtexfile.bib
+++ b/jump_lin_id/bibtexfile.bib
@@ -171,3 +171,25 @@
XXpages={1445--1450},
year={1965}
}
+
+@ARTICLE{svensson2014identification,
+ author = {{Svensson}, A. and {Sch{\"o}n}, T.~B. and {Lindsten}, F.},
+ title = "{Identification of jump Markov linear models using particle filters}",
+ journal = {ArXiv e-prints},
+archivePrefix = "arXiv",
+ eprint = {1409.7287},
+ primaryClass = "stat.CO",
+ keywords = {Statistics - Computation, Mathematics - Optimization and Control, Statistics - Machine Learning},
+ year = 2014,
+ month = sep,
+ adsurl = {http://adsabs.harvard.edu/abs/2014arXiv1409.7287S},
+ adsnote = {Provided by the SAO/NASA Astrophysics Data System}
+}
+
+@book{gustafsson2000adaptive,
+ title={Adaptive filtering and change detection},
+ author={Gustafsson, Fredrik and Gustafsson, Fredrik},
+ volume={1},
+ year={2000},
+ publisher={Citeseer}
+}
diff --git a/jump_lin_id/pres/beamerthemeRegler2.sty b/jump_lin_id/pres/beamerthemeRegler2.sty
index 2147726..8d2bd77 100644
--- a/jump_lin_id/pres/beamerthemeRegler2.sty
+++ b/jump_lin_id/pres/beamerthemeRegler2.sty
@@ -1,45 +1,3 @@
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- height\ht\beamer@linebox\hskip-\paperwidth%
- \hbox{\box\beamer@linebox\hfill}\hfill\hskip-\Gm@rmargin}}}
-
-\setbeamercovered{transparent}
-
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-
\DeclareOption{liontopcorner}{\def\@beamer@option{%
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\useframetitletemplate{\par\kern-1mm
@@ -59,9 +17,9 @@
\hbox{%
\setbox\beamer@linebox=\hbox to\paperwidth{%
\hbox to.5\paperwidth{\tiny\color{white}\frame@numbers
- \hfill\textbf{\insertshortauthor}\hskip.3cm}%
+ \hfill\textbf{\shadowtext{\insertshortauthor}}\hskip.3cm}%
\hbox to.5\paperwidth{\hskip.3cm\tiny\color{white}%
- \textbf{\insertshorttitle}\hfill}\hfill}%
+ \textbf{\shadowtext{\insertshorttitle \quad \insertshortdate}}\hfill}\hfill}%
\ht\beamer@linebox=2.625ex%
\dp\beamer@linebox=0pt%
\setbox\beamer@linebox=\vbox{\box\beamer@linebox\vskip1.125ex}%
@@ -74,118 +32,6 @@
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-\newsavebox\lioncornerbox
-\AtBeginDocument {%
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\DeclareOption{handout}{\def\@beamer@option{%
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@@ -242,7 +88,9 @@
\vspace{5mm}
{\normalsize\textbf\insertauthor\par}
\vspace{5mm}
- {\scriptsize\insertinstitute\par}%\par\vskip1em
+ {\scriptsize\insertinstitute\par}
+ \vspace{5mm}
+ {\scriptsize\insertdate\par}%\par\vskip1em
\end{centering}
\vss
}
diff --git a/jump_lin_id/pres/pres_idpaper.tex b/jump_lin_id/pres/pres_idpaper.tex
index a996519..dfae9ee 100644
--- a/jump_lin_id/pres/pres_idpaper.tex
+++ b/jump_lin_id/pres/pres_idpaper.tex
@@ -1,4 +1,4 @@
-\documentclass[10pt,handout]{beamer}
+\documentclass[10pt]{beamer}
% \usepackage{pgfpages} \pgfpagesuselayout{8 on 1}[a4paper,border shrink=5mm]
\usetheme[liontopcorner,framenumbers]{Regler2}
\usepackage{graphicx}
@@ -14,6 +14,7 @@
\addbibresource{../bibtexfile.bib}
\usepackage{siunitx}
\usepackage{color}
+\usepackage{shadowtext}\shadowoffset{0.05mm}\shadowcolor{black!80!white}
\usepackage{pgfplots}
\usepackage{booktabs}\usepackage{multirow}
\usepgfplotslibrary{groupplots}
@@ -48,9 +49,9 @@ label=center:{{$\sum$}}, minimum width=2em}}
\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}}
\institute{Lund University, Department of Automatic Control}
+\date[]{ICCA Anchorage, June 2018}
\definecolor{red}{rgb}{0.7,0.2,0.2}
@@ -93,29 +94,64 @@ label=center:{{$\sum$}}, minimum width=2em}}
\newcommand{\bmatrixx}[1]{\begin{bmatrix}#1\end{bmatrix}}
+\usepgfplotslibrary{external}
+\tikzexternalize
+\tikzsetexternalprefix{figs/}
\begin{document}
+\setbeamercolor{footnote}{bg=mintgreen}
\newlength\figureheight
\newlength\figurewidth
% \setbeamercolor{background canvas}{bg=goldishlight}
\maketitle
+%====================================================================
+%====================================================================
+\begin{frame}{Slides and Code}{}
+ Slides, code, figures and examples available at
+
+ \centering
+ \includegraphics[width=0.3\linewidth]{figs/qr.png}
+
+ \url{github.com/baggepinnen/LTVModels.jl}
+\end{frame}
+
+
+%====================================================================
+%====================================================================
+\begin{frame}{Outline}{}
+ \begin{enumerate}
+ \item LTI identification
+ \item LTV identification (Main topic)
+ \item Examples
+ \end{enumerate}
+\end{frame}
+
%====================================================================
%====================================================================
\begin{frame}{LTI identification}
- We start by considering the case of identification of the parameters in an LTI model on the form
+ LTI model
\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.
+ \pause
+
+ \begin{itemize}
+ \item Control design
+ \item Prediction
+ \item Simulation
+ \end{itemize}
\end{frame}
-\begin{frame}{}{}
- $\y = \A\w$, and arrange the data according to
+
+
+\begin{frame}{LTI identification}{}
+ Linear in the parameters, can be written $\y = \A\w$
\begin{align*}
\y &=
\begin{bmatrix}
@@ -131,9 +167,11 @@ label=center:{{$\sum$}}, minimum width=2em}}
\end{bmatrix}
& &\in \mathbb{R}^{Tn\times K}
\end{align*}
+ \pause
+ Closed form solution
\begin{align}
- \w^* &= \argmin_{\w} \normt{\A \w - \y}^2 \label{eq:lscost}\\
+ \w^* &= \argmin_{\w} \normt{\y - \A \w}^2 \label{eq:lscost}\\
~ &= \PI \y \label{eq:ls}
\end{align}
@@ -144,17 +182,20 @@ label=center:{{$\sum$}}, minimum width=2em}}
%====================================================================
%====================================================================
-\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}
+\begin{frame}{Problem formulation}
+ Estimate a model on the form
+ \begin{block}{Linear Time-Varying (LTV) dynamics}
+ \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}
+ \end{block}
\pause
- where the parameters $\w$ are assumed to evolve according to the dynamical system
+
+ Parameters $\w$ are assumed to evolve according to the dynamical system
\begin{equation}
\label{eq:dynsys}
\begin{split}
@@ -162,6 +203,28 @@ label=center:{{$\sum$}}, minimum width=2em}}
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}
+
+
+%====================================================================
+%====================================================================
+\begin{frame}{Identification of LTV models}{}
+ \begin{block}{Previous research}
+ \begin{itemize}
+ \item Kalman filter with restarts\footfullcite{gustafsson2000adaptive}
+ \item Segmented least-squares\footfullcite{bellman1969curve}
+ \item EM and particle filtering\footfullcite{svensson2014identification}
+ \end{itemize}
+ \end{block}
+ \pause
+
+ \begin{itemize}
+ \item Review of trend filtering
+ \item Review of regularization properties of norms
+ \end{itemize}
\end{frame}
@@ -170,12 +233,14 @@ label=center:{{$\sum$}}, minimum width=2em}}
%====================================================================
%====================================================================
\begin{frame}{Trend filtering}
- An important class of identification methods that has been popularized lately is \emph{trend filtering} methods~\footfullcite{kim2009ell_1, tibshirani2014adaptive}.
+ Class of identification methods for signal reconstruction, \emph{trend filtering}~\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{itemize}
\item Fitness function
\item (Sparsity promoting) Regularization
@@ -186,29 +251,48 @@ label=center:{{$\sum$}}, minimum width=2em}}
%====================================================================
%====================================================================
-\begin{frame}{Regularization term intuition}{}
- figure
+\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.
+ \end{description}
- 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}{Identification methods}{}
+ \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
+ \end{itemize}
+\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}
\pause
+ Closed-form solution available \only<4->{\bad{(intractable)}}
\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}
+ \pause
+
+ Parameters found efficiently by Dynamic programming!
\end{frame}
@@ -220,9 +304,16 @@ label=center:{{$\sum$}}, minimum width=2em}}
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}
@@ -234,7 +325,7 @@ label=center:{{$\sum$}}, minimum width=2em}}
\begin{equation} \label{eq:pwconstant}
- \minimize{\w} \normt{\y-\hat{\y}}^2 + \lambda\sum_t \normt{ \w_{t+1} - \w_t}
+ \minimize{\w} \normt{\y-\hat{\y}}^2 + \lambda\sum_t \normt{ \w_{t+1} - \w_t}^{\only<3>{\bad{2}}}
\end{equation}
\pause
@@ -245,7 +336,7 @@ label=center:{{$\sum$}}, minimum width=2em}}
\end{frame}
-\begin{frame}{}{}
+\begin{frame}{Piecewise constant time evolution}{}
At a first glance, one might consider the formulation
\begin{equation} \label{eq:pwconstant_naive}
@@ -253,14 +344,14 @@ label=center:{{$\sum$}}, minimum width=2em}}
\end{equation}
\pause
- changes to different entries of $\w_t$ would not occur at the same time instants.
+ 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}.
+\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}.
An efficient implementation using the linearized ADMM algorithm \footfullcite{parikh2014proximal} is made available in the accompanying repository.
@@ -272,11 +363,9 @@ label=center:{{$\sum$}}, minimum width=2em}}
%====================================================================
%====================================================================
\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 $n$.}
+ \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
@@ -297,7 +386,9 @@ label=center:{{$\sum$}}, minimum width=2em}}
%====================================================================
%====================================================================
\begin{frame}{Example -- Jump-linear system}
- We now consider a simulated example. Change in dynamics, from
+ We now consider a simulated example.
+
+ Change in dynamics, from
$$A_t = \left[
\begin{array}{cc}
0.95 & 0.1 \\
@@ -326,7 +417,7 @@ label=center:{{$\sum$}}, minimum width=2em}}
\begin{description}
\item[Input] $u \sim \N(0, 1)$
- \item[state transition noise and measurement noise] $\N(0, 0.2^2)$
+ \item[State transition / measurement noise] $\N(0, 0.2^2)$
\end{description}
\end{frame}
@@ -341,6 +432,9 @@ label=center:{{$\sum$}}, minimum width=2em}}
\label{fig:ss}
\end{figure}
+ {\color{gray}
+ Code to reproduce: \href{https://github.com/baggepinnen/LTVModels.jl}{github.com/baggepinnen/LTVModels.jl}
+ }
\end{frame}
@@ -348,11 +442,29 @@ label=center:{{$\sum$}}, minimum width=2em}}
%====================================================================
%====================================================================
\begin{frame}{Example -- Robot arm}
- \begin{itemize}
- \item Non-smooth dynamics
- \item Discontinuous Coulomb friction
- \item Stiff contact with environment
- \end{itemize}
+
+ \begin{columns}
+ \begin{column}{0.5\textwidth}
+ \begin{itemize}
+ \item Non-smooth dynamics
+ \item Discontinuous Coulomb friction
+ \item Stiff contact with environment
+ \item Learn LTV model for trajectory optimization
+ \item Reinforcement learning
+ \end{itemize}
+ \end{column}
+ \begin{column}{0.5\textwidth}
+ \centering
+ \includegraphics[width=\linewidth]{figs/robot_draw.jpg}
+ % \input{figs/robot.tex}
+ \end{column}
+ \end{columns}
+
+ \vspace{1cm}
+ {\color{gray}
+ Code to reproduce: \href{https://github.com/baggepinnen/LTVModels.jl/blob/master/examples/two_link.jl}{github.com/baggepinnen/LTVModels.jl/blob/master/examples/two\_link.jl}}
+
+
% 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.
@@ -360,7 +472,7 @@ label=center:{{$\sum$}}, minimum width=2em}}
%====================================================================
%====================================================================
\begin{frame}{Robot -- Training trajectory}{}
-
+ \vspace*{-6mm}
\begin{figure}
\centering
\pgfplotsset{every axis/.append style={
@@ -377,17 +489,18 @@ label=center:{{$\sum$}}, minimum width=2em}}
%====================================================================
%====================================================================
\begin{frame}{Robot -- Validation trajectory}{}
+ \vspace*{-6mm}
\begin{figure}
\centering
\pgfplotsset{every axis/.append style={
- label style={font=\tiny},
- legend style={font=\tiny, draw=none},
- tick label style={font=\tiny}
+ label style={font=\scriptsize},
+ legend style={font=\scriptsize, draw=none},
+ tick label style={font=\scriptsize}
}}
\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.}
+ \caption{Validation data vs. sample time index. The dashed lines indicate the event times for the training data.}
\label{fig:robot_val}
\end{figure}
\end{frame}
@@ -398,40 +511,49 @@ label=center:{{$\sum$}}, minimum width=2em}}
%====================================================================
\begin{frame}{Example -- Reinforcement learning} \label{sec:rl}
\begin{itemize}
- \item Identify LTV dynamics models for reinforcement learning
+ \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 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
+ \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 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.
+ \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}
- 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}
@@ -451,9 +573,10 @@ label=center:{{$\sum$}}, minimum width=2em}}
%====================================================================
\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.
+ \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}. \\
+ Can lead to dramatically decreased sample complexity.
\end{itemize}
\end{frame}
@@ -464,7 +587,12 @@ label=center:{{$\sum$}}, minimum width=2em}}
\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
+ \item Applications within
+ \begin{itemize}
+ \item Trajectory-centric, model-based reinforcement learning, ILC
+ \item Jump-linear system identification
+ \item Change-point detection
+ \end{itemize}
\end{itemize}
\pause
@@ -479,8 +607,13 @@ label=center:{{$\sum$}}, minimum width=2em}}
%====================================================================
%====================================================================
\begin{frame}{Open source}{}
- Code to train the models and reproduce examples presented in this talk available at
- \url{https://github.com/baggepinnen/LTVModels.jl}
+ More examples and code to train the models and reproduce examples presented in this talk available at
+
+ \centering
+ \url{github.com/baggepinnen/LTVModels.jl}
+
+ \includegraphics[width=0.4\linewidth]{figs/qr.png}
+
\end{frame}
--
GitLab