[ellipsoids] r2764 committed - issue 133...

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Revision: 2764
Author: anna.ata...@gmail.com
Date: Thu Mar 6 15:09:17 2014 UTC
Log: issue 133
Enhancement: committing a few changes
http://code.google.com/p/ellipsoids/source/detail?r=2764

Modified:
/branches/issue_133_aatanesyan/doc/_build/html/.buildinfo
/branches/issue_133_aatanesyan/doc/_build/html/_sources/chap_ellTube.txt
/branches/issue_133_aatanesyan/doc/_build/html/_sources/chap_implement.txt
/branches/issue_133_aatanesyan/doc/_build/html/chap_acknowledge.html
/branches/issue_133_aatanesyan/doc/_build/html/chap_ellTube.html
/branches/issue_133_aatanesyan/doc/_build/html/chap_ellcalc.html
/branches/issue_133_aatanesyan/doc/_build/html/chap_examples.html
/branches/issue_133_aatanesyan/doc/_build/html/chap_functions.html
/branches/issue_133_aatanesyan/doc/_build/html/chap_implement.html
/branches/issue_133_aatanesyan/doc/_build/html/chap_install.html
/branches/issue_133_aatanesyan/doc/_build/html/chap_intro.html
/branches/issue_133_aatanesyan/doc/_build/html/chap_reach.html
/branches/issue_133_aatanesyan/doc/_build/html/chap_summary.html
/branches/issue_133_aatanesyan/doc/_build/html/genindex.html
/branches/issue_133_aatanesyan/doc/_build/html/main_manual.html
/branches/issue_133_aatanesyan/doc/_build/html/objects.inv
/branches/issue_133_aatanesyan/doc/_build/html/search.html
/branches/issue_133_aatanesyan/doc/_build/html/searchindex.js
/branches/issue_133_aatanesyan/doc/_build/latex/elltool_manual.tex
/branches/issue_133_aatanesyan/doc/chap_ellTube.rst
/branches/issue_133_aatanesyan/doc/chap_implement.rst
/branches/issue_133_aatanesyan/doc/conf.py
/branches/issue_133_aatanesyan/doc/main_elltoolmanual.tex

=======================================
--- /branches/issue_133_aatanesyan/doc/_build/html/.buildinfo Mon Feb 24
20:40:31 2014 UTC
+++ /branches/issue_133_aatanesyan/doc/_build/html/.buildinfo Thu Mar 6
15:09:17 2014 UTC
@@ -1,4 +1,4 @@
# Sphinx build info version 1
# This file hashes the configuration used when building these files. When
it is not found, a full rebuild will be done.
-config: 982ebbdc8f2d8f8beaef0b683df722af
+config: 9cb5c1e55f016cf97bb882dc74d49254
tags: 645f666f9bcd5a90fca523b33c5a78b7
=======================================
---
/branches/issue_133_aatanesyan/doc/_build/html/_sources/chap_ellTube.txt
Wed Mar 5 22:30:52 2014 UTC
+++
/branches/issue_133_aatanesyan/doc/_build/html/_sources/chap_ellTube.txt
Thu Mar 6 15:09:17 2014 UTC
@@ -1,35 +1,37 @@
-Ellipsoid tubes, tubes by the instant of time and their projections
+Ellipsoid tubes, tubes by instant of time and their projections
===================================================================

-**Definition.** For any matrix asimptotic monotone
function :math:`M(\cdot)`, which defines the configuration of the
environment of any point of time, the quadratically regularized alternated
reach set of system with disturbance is
+**Definition.** For any regularization matrix :math:`M(\cdot)` the
quadratically regularized alternated reach set of system with uncertainty is

.. math::
{\mathcal X}^{q}_{U}(t,t_0,{\mathcal X}^{0},M(\cdot)) =
\underset{t_0\leq\tau\leq t}{\bigcup}{\mathcal
X}^{q}_{U}(\tau,t_0,{\mathcal X}^{0},M (\cdot)).

-Identify as :math:`{\mathcal E}(\overline{x}(t),X_{+}(t,l))`
и :math:`{\mathcal E}(\overline{x}(t),X_{-}(t,l))` the tight external and
internal approximations along :math:`l(\cdot)` good direction such
as :math:`l(t_0)=l`. Then the reach set by instant of time can be described
as
+Note that if system doesn't have uncertainty then :math:`M(\cdot)=0`.
+
+By :math:`{\mathcal E}(\overline{x}(t),X_{+}(t,l))` and :math:`{\mathcal
E}(\overline{x}(t),X_{-}(t,l))` denote tight external and internal
approximations along :math:`l(\cdot)` good direction such
as :math:`l(t_0)=l`. Then the reach set by instant of time can be described
as

.. math::
{\mathcal X}^{q}_{U}[t]=\underset{\tau}{\bigcup}\underset{l}{\bigcap}\{
{\mathcal E}(\overline{x}(\tau),X_{+}(\tau,l)) |l\in {\mathcal
S}_1(0), \tau\leq t\}\subseteq\underset{l} {\bigcap}\{{\mathcal
E}^{U}_{+}[t,l]) |l\in {\mathcal S}_1(0)\},

-where :math:`{\mathcal E}^{U}_{+}[t,l]=\underset{\tau}{\bigcup}\{{\mathcal
E}(\overline{x}(\tau),X_{+}(\tau,l))|t_0\leq\tau\leq t\}` is the external
ellipsoidal tube by the instant of time.
+where :math:`{\mathcal E}^{U}_{+}[t,l]=\underset{\tau}{\bigcup}\{{\mathcal
E}(\overline{x}(\tau),X_{+}(\tau,l))|t_0\leq\tau\leq t\}` is the external
ellipsoidal tube by instant of time.

-Similar approxiamtion can be calculated with the internal ellipsoidal tube
by the instant of time :math:`{\mathcal
E}^{U}_{-}[t,l]=\underset{\tau}{\bigcup}\{{\mathcal
E}(\overline{x}(\tau),X_{-}(\tau,l))|t_0\leq\tau\leq t\}`:
+Similar approxiamtion can be calculated with the internal ellipsoidal tube
by instant of time :math:`{\mathcal
E}^{U}_{-}[t,l]=\underset{\tau}{\bigcup}\{{\mathcal
E}(\overline{x}(\tau),X_{-}(\tau,l))|t_0\leq\tau\leq t\}`:

.. math::
{\mathcal X}^{U}[t]\supseteq\underset{l}{\bigcup}\{{\mathcal
E}^{U}_{-}[t,l]) |l\in {\mathcal S}_1(0)\}.

-Note that in general case ellipsoidal tube :math:`{\mathcal
E}^{U}_{+}[t,l]` is not tight approximation.
+Note that in general case ellipsoidal tube :math:`{\mathcal
E}^{U}_{+}[t,l]` is not tight approximation. For more information see
[GAG2012]_.

So, all in all, there are two types of ellipsoid tube objects that we can
work with using *Ellipsoidal Toolbox*:

- ellipsoidal tubes that are described in
*gras.ellapx.smartdb.rels.EllTube* class;

-- tubes by the instant of time described in
*gras.ellapx.smartdb.rels.EllUnionTube* class
+- tubes by instant of time described in
*gras.ellapx.smartdb.rels.EllUnionTube* class
(see :ref:`formula <union-label>`).

-These two type of objects can be projected on specified subspaces. The
projections of ellipsoid tubes can be either static or dynamic. The are
described in gras.ellapx.smartdb.rels.EllTubeProj class. As for the tubes
by the instant of time, they can only be projected on static subspaces.
These projections are described in
gras.ellapx.smartdb.rels.EllUnionTubeStaticProj class. For more information
about these types of projections and their differences and for examples see
this :ref:`link <section-label>`.
+These two type of objects can be projected on specified subspaces. The
projections of ellipsoid tubes can be either static or dynamic. The are
described in gras.ellapx.smartdb.rels.EllTubeProj class. As for the tubes
by instant of time, they can only be projected on static subspaces. These
projections are described in
gras.ellapx.smartdb.rels.EllUnionTubeStaticProj class. For more information
about these types of projections and their differences and for examples see
this :ref:`link <section-label>`.

Ellipsoid tubes
---------------
@@ -195,16 +197,16 @@
:language: matlab
:linenos:

-Tubes by the instant of time
-----------------------------
+Tubes by instant of time
+------------------------

-As with ellipsoid tube objects there are several methods that we can use
while working with tubes by the instant of time. First of all we can create
tubes by the instant of time using *fromEllTubes* method:
+As with ellipsoid tube objects, there are several methods that we can use
while working with tubes by instant of time. First of all we can create
tubes by instant of time using *fromEllTubes* method:

..
literalinclude:: ../products/+gras/+ellapx/+smartdb/+test/+examples//example_fromEllTubes.m
:language: matlab
:linenos:

-From here on we will use the *getUnion* function so we can get a tube by
the instant of time and work with it further on. As we have created a tubes
by the instant of time object, we can get all the types of differet data
about it. There is a set of methods that can give information about the
data stored in the object and give access to it.
+From here on we will use the *getUnion* function so we can get a tube by
instant of time and work with it further on. As we have created a tubes by
instant of time object, we can get all the types of differet data about it.
There is a set of methods that can give information about the data stored
in the object and give access to it.

..
literalinclude:: ../products/+gras/+ellapx/+smartdb/+test/+examples//example_getDataUnion.m
:language: matlab
@@ -225,13 +227,13 @@
:language: matlab
:linenos:

-Also we can compare tubes by the instant of time using *isEqual* method.
+Also we can compare tubes by instant of time using *isEqual* method.

..
literalinclude:: ../products/+gras/+ellapx/+smartdb/+test/+examples//example_isEqualUnion.m
:language: matlab
:linenos:

-At last, as it has already been said tubes by the instant of time can be
projected only on static subspaces. It can be done in two ways.
+At last, as it has already been said tubes by instant of time can be
projected only on static subspaces. It can be done in two ways.

..
literalinclude:: ../products/+gras/+ellapx/+smartdb/+test/+examples//example_project.m
:language: matlab
@@ -241,28 +243,28 @@
:language: matlab
:linenos:

-As for ellipsoid tube projections, a special function is used to create
the projection matrix for tubes by the instant of time:
+As for ellipsoid tube projections, a special function is used to create
the projection matrix for tubes by instant of time:

..
literalinclude:: ../products/+gras/+ellapx/+smartdb/+test/+examples//fGetProjMat.m
:language: matlab
:linenos:

-Projections of ellipsoid tubes and unions
------------------------------------------
+Projections of ellipsoid tubes and tubes by instant of time
+-----------------------------------------------------------

-As it has already been said we can create either static or dynamic
projections for ellipsoid tubes and only static projections for tubes by
the instant of time. There are several methods in *Ellipsoidal Toolbox* for
that. Most of them has already been described:
+As it has already been said we can create either static or dynamic
projections for ellipsoid tubes and only static projections for tubes by
instant of time. There are several methods in *Ellipsoidal Toolbox* for
that. Most of them has already been described:

- *project*, *projectStatic* and *projectToOrths* for ellipsoid tubes;

-- *project* and *projectStatic* for tubes by the instant of time.
+- *project* and *projectStatic* for tubes by instant of time.

-It should be mentioned that from here on all the examples are written for
ellipsoid tube projections, but their usage is the same for the projections
of tubes by the instant of time. We wiil use *getProj* function to create
ellipsoid tube projection that we will work with.
+It should be mentioned that from here on all the examples are written for
ellipsoid tube projections, but their usage is the same for the projections
of tubes by instant of time. We wiil use *getProj* function to create
ellipsoid tube projection that we will work with.

..
literalinclude:: ../products/+gras/+ellapx/+smartdb/+test/+examples//example_getProj.m
:language: matlab
:linenos:

-As with ellipsoid tubes and tubes by the instant of time we can get all
the types of differet data about projections. There is a set of methods
that can give information about the data stored in the object and give
access to it.
+As with ellipsoid tubes and tubes by instant of time we can get all the
types of differet data about projections. There is a set of methods that
can give information about the data stored in the object and give access to
it.

..
literalinclude:: ../products/+gras/+ellapx/+smartdb/+test/+examples//example_getDataProj.m
@@ -307,3 +309,4 @@

To read more about the differences of these projections :ref:`goto
<goto-label>`. Also there you can find some illustrations for both ellTube
projections and ellUnionTube projections.

+.. [GAG2012] Gagarinov P.V. 2012. Computation of Alternated Reachability
Tubes for Linear Control Systems under Uncertainty. *Moscow University
Computational Mathematics and Cybernetics*, 2012, Vol. 36, No. 4, pp. 169–
177.
=======================================
---
/branches/issue_133_aatanesyan/doc/_build/html/_sources/chap_implement.txt
Wed Mar 5 02:41:50 2014 UTC
+++
/branches/issue_133_aatanesyan/doc/_build/html/_sources/chap_implement.txt
Thu Mar 6 15:09:17 2014 UTC
@@ -405,6 +405,7 @@
.. _union-label:

.. math::
+ :label: ellUnion

{\mathcal X}_{U}[t]=\bigcup \limits_{\tau\leqslant t}{\mathcal X}[\tau],

=======================================
--- /branches/issue_133_aatanesyan/doc/_build/html/chap_acknowledge.html
Mon Feb 24 20:40:31 2014 UTC
+++ /branches/issue_133_aatanesyan/doc/_build/html/chap_acknowledge.html
Thu Mar 6 15:09:17 2014 UTC
@@ -6,7 +6,7 @@
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />

- <title>Acknowledgement — Ellipsoidal Toolbox 2.0 beta 1
documentation</title>
+ <title>Acknowledgement — Ellipsoidal Toolbox 2.0.1
documentation</title>

<link rel="stylesheet" href="_static/default.css" type="text/css" />
<link rel="stylesheet" href="_static/pygments.css" type="text/css" />
@@ -14,7 +14,7 @@
<script type="text/javascript">
var DOCUMENTATION_OPTIONS = {
URL_ROOT: './',
- VERSION: '2.0 beta 1',
+ VERSION: '2.0.1',
COLLAPSE_INDEX: false,
FILE_SUFFIX: '.html',
HAS_SOURCE: true
@@ -23,7 +23,7 @@
<script type="text/javascript" src="_static/jquery.js"></script>
<script type="text/javascript" src="_static/underscore.js"></script>
<script type="text/javascript" src="_static/doctools.js"></script>
- <link rel="top" title="Ellipsoidal Toolbox 2.0 beta 1 documentation"
href="index.html" />
+ <link rel="top" title="Ellipsoidal Toolbox 2.0.1 documentation"
href="index.html" />
<link rel="next" title="Function Reference" href="chap_functions.html"
/>
<link rel="prev" title="Summary and Outlook" href="chap_summary.html"
/>
</head>
@@ -40,7 +40,7 @@
<li class="right" >
<a href="chap_summary.html" title="Summary and Outlook"
accesskey="P">previous</a> |</li>
- <li><a href="main_manual.html">Ellipsoidal Toolbox 2.0 beta 1
documentation</a> »</li>
+ <li><a href="main_manual.html">Ellipsoidal Toolbox 2.0.1
documentation</a> »</li>
</ul>
</div>

@@ -102,7 +102,7 @@
<li class="right" >
<a href="chap_summary.html" title="Summary and Outlook"
>previous</a> |</li>
- <li><a href="main_manual.html">Ellipsoidal Toolbox 2.0 beta 1
documentation</a> »</li>
+ <li><a href="main_manual.html">Ellipsoidal Toolbox 2.0.1
documentation</a> »</li>
</ul>
</div>
<div class="footer">
=======================================
--- /branches/issue_133_aatanesyan/doc/_build/html/chap_ellTube.html Wed
Mar 5 22:30:52 2014 UTC
+++ /branches/issue_133_aatanesyan/doc/_build/html/chap_ellTube.html Thu
Mar 6 15:09:17 2014 UTC
@@ -6,7 +6,7 @@
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />

- <title>Ellipsoid tubes, tubes by the instant of time and their
projections — Ellipsoidal Toolbox 2.0 beta 1 documentation</title>
+ <title>Ellipsoid tubes, tubes by instant of time and their projections
— Ellipsoidal Toolbox 2.0.1 documentation</title>

<link rel="stylesheet" href="_static/default.css" type="text/css" />
<link rel="stylesheet" href="_static/pygments.css" type="text/css" />
@@ -14,7 +14,7 @@
<script type="text/javascript">
var DOCUMENTATION_OPTIONS = {
URL_ROOT: './',
- VERSION: '2.0 beta 1',
+ VERSION: '2.0.1',
COLLAPSE_INDEX: false,
FILE_SUFFIX: '.html',
HAS_SOURCE: true
@@ -23,7 +23,7 @@
<script type="text/javascript" src="_static/jquery.js"></script>
<script type="text/javascript" src="_static/underscore.js"></script>
<script type="text/javascript" src="_static/doctools.js"></script>
- <link rel="top" title="Ellipsoidal Toolbox 2.0 beta 1 documentation"
href="index.html" />
+ <link rel="top" title="Ellipsoidal Toolbox 2.0.1 documentation"
href="index.html" />
<link rel="next" title="Summary and Outlook" href="chap_summary.html"
/>
<link rel="prev" title="Examples" href="chap_examples.html" />
</head>
@@ -40,7 +40,7 @@
<li class="right" >
<a href="chap_examples.html" title="Examples"
accesskey="P">previous</a> |</li>
- <li><a href="main_manual.html">Ellipsoidal Toolbox 2.0 beta 1
documentation</a> »</li>
+ <li><a href="main_manual.html">Ellipsoidal Toolbox 2.0.1
documentation</a> »</li>
</ul>
</div>

@@ -49,28 +49,29 @@
<div class="bodywrapper">
<div class="body">

- <div class="section"
id="ellipsoid-tubes-tubes-by-the-instant-of-time-and-their-projections">
-<h1>Ellipsoid tubes, tubes by the instant of time and their projections<a
class="headerlink"
href="#ellipsoid-tubes-tubes-by-the-instant-of-time-and-their-projections"
title="Permalink to this headline">¶</a></h1>
-<p><strong>Definition.</strong> For any matrix asimptotic monotone
function <img class="math"
src="_images/math/2e573bfe03a50bdd1738c6cf4dc80d417afd1c05.png"
alt="M(\cdot)"/>, which defines the configuration of the environment of any
point of time, the quadratically regularized alternated reach set of system
with disturbance is</p>
+ <div class="section"
id="ellipsoid-tubes-tubes-by-instant-of-time-and-their-projections">
+<h1>Ellipsoid tubes, tubes by instant of time and their projections<a
class="headerlink"
href="#ellipsoid-tubes-tubes-by-instant-of-time-and-their-projections"
title="Permalink to this headline">¶</a></h1>
+<p><strong>Definition.</strong> For any regularization matrix <img
class="math"
src="_images/math/2e573bfe03a50bdd1738c6cf4dc80d417afd1c05.png"
alt="M(\cdot)"/> the quadratically regularized alternated reach set of
system with uncertainty is</p>
<div class="math">
<p><img src="_images/math/512c3855253a9608744f87b71998cc4099031188.png"
alt="{\mathcal X}^{q}_{U}(t,t_0,{\mathcal X}^{0},M(\cdot)) =
\underset{t_0\leq\tau\leq t}{\bigcup}{\mathcal
X}^{q}_{U}(\tau,t_0,{\mathcal X}^{0},M (\cdot))."/></p>
-</div><p>Identify as <img class="math"
src="_images/math/06b60d0d7214d59442d08b7e393336ceb6c7f87e.png"
alt="{\mathcal E}(\overline{x}(t),X_{+}(t,l))"/> и <img class="math"
src="_images/math/8186d35faf6b4d13f95ad584cf4f1ac0b930749d.png"
alt="{\mathcal E}(\overline{x}(t),X_{-}(t,l))"/> the tight external and
internal approximations along <img class="math"
src="_images/math/28c7cde33aa87f676e28cc8f7c2c65e61783f595.png"
alt="l(\cdot)"/> good direction such as <img class="math"
src="_images/math/0c7534cf3930d33d9e93b380d3282a37068e1ce1.png"
alt="l(t_0)=l"/>. Then the reach set by instant of time can be described
as</p>
+</div><p>Note that if system doesn’t have uncertainty then <img
class="math"
src="_images/math/0ab9370bf1570c1658bf09d2d6ff6a8ecda75db8.png"
alt="M(\cdot)=0"/>.</p>
+<p>By <img class="math"
src="_images/math/06b60d0d7214d59442d08b7e393336ceb6c7f87e.png"
alt="{\mathcal E}(\overline{x}(t),X_{+}(t,l))"/> and <img class="math"
src="_images/math/8186d35faf6b4d13f95ad584cf4f1ac0b930749d.png"
alt="{\mathcal E}(\overline{x}(t),X_{-}(t,l))"/> denote tight external and
internal approximations along <img class="math"
src="_images/math/28c7cde33aa87f676e28cc8f7c2c65e61783f595.png"
alt="l(\cdot)"/> good direction such as <img class="math"
src="_images/math/0c7534cf3930d33d9e93b380d3282a37068e1ce1.png"
alt="l(t_0)=l"/>. Then the reach set by instant of time can be described
as</p>
<div class="math">
<p><img src="_images/math/957b5c775a16f08451308745dc2dae9e86f82ee9.png"
alt="{\mathcal X}^{q}_{U}[t]=\underset{\tau}{\bigcup}\underset{l}{\bigcap}\{
{\mathcal E}(\overline{x}(\tau),X_{+}(\tau,l)) |l\in {\mathcal S}_1(0),
\tau\leq t\}\subseteq\underset{l} {\bigcap}\{{\mathcal E}^{U}_{+}[t,l]) |
l\in {\mathcal S}_1(0)\},"/></p>
-</div><p>where <img class="math"
src="_images/math/cca4a283614b3ed693d0ae8c1a86c2486bf69dbb.png"
alt="{\mathcal E}^{U}_{+}[t,l]=\underset{\tau}{\bigcup}\{{\mathcal
E}(\overline{x}(\tau),X_{+}(\tau,l))|t_0\leq\tau\leq t\}"/> is the external
ellipsoidal tube by the instant of time.</p>
-<p>Similar approxiamtion can be calculated with the internal ellipsoidal
tube by the instant of time <img class="math"
src="_images/math/1798176817126eb51cd40d65d0242caea8004c97.png"
alt="{\mathcal E}^{U}_{-}[t,l]=\underset{\tau}{\bigcup}\{{\mathcal
E}(\overline{x}(\tau),X_{-}(\tau,l))|t_0\leq\tau\leq t\}"/>:</p>
+</div><p>where <img class="math"
src="_images/math/cca4a283614b3ed693d0ae8c1a86c2486bf69dbb.png"
alt="{\mathcal E}^{U}_{+}[t,l]=\underset{\tau}{\bigcup}\{{\mathcal
E}(\overline{x}(\tau),X_{+}(\tau,l))|t_0\leq\tau\leq t\}"/> is the external
ellipsoidal tube by instant of time.</p>
+<p>Similar approxiamtion can be calculated with the internal ellipsoidal
tube by instant of time <img class="math"
src="_images/math/1798176817126eb51cd40d65d0242caea8004c97.png"
alt="{\mathcal E}^{U}_{-}[t,l]=\underset{\tau}{\bigcup}\{{\mathcal
E}(\overline{x}(\tau),X_{-}(\tau,l))|t_0\leq\tau\leq t\}"/>:</p>
<div class="math">
<p><img src="_images/math/1b65b4393db2e5450f682d2d746f10f530d5f150.png"
alt="{\mathcal X}^{U}[t]\supseteq\underset{l}{\bigcup}\{{\mathcal
E}^{U}_{-}[t,l]) |l\in {\mathcal S}_1(0)\}."/></p>
-</div><p>Note that in general case ellipsoidal tube <img class="math"
src="_images/math/d9a7eb7a9cae035a5c7359576b38ab00c902054e.png"
alt="{\mathcal E}^{U}_{+}[t,l]"/> is not tight approximation.</p>
+</div><p>Note that in general case ellipsoidal tube <img class="math"
src="_images/math/d9a7eb7a9cae035a5c7359576b38ab00c902054e.png"
alt="{\mathcal E}^{U}_{+}[t,l]"/> is not tight approximation. For more
information see <a class="reference internal" href="#gag2012"
id="id1">[GAG2012]</a>.</p>
<p>So, all in all, there are two types of ellipsoid tube objects that we
can work with using <em>Ellipsoidal Toolbox</em>:</p>
<ul class="simple">
<li>ellipsoidal tubes that are described in
<em>gras.ellapx.smartdb.rels.EllTube</em> class;</li>
-<li>tubes by the instant of time described in
<em>gras.ellapx.smartdb.rels.EllUnionTube</em> class
+<li>tubes by instant of time described in
<em>gras.ellapx.smartdb.rels.EllUnionTube</em> class
(see <a class="reference internal"
href="chap_implement.html#union-label"><em>formula</em></a>).</li>
</ul>
-<p>These two type of objects can be projected on specified subspaces. The
projections of ellipsoid tubes can be either static or dynamic. The are
described in gras.ellapx.smartdb.rels.EllTubeProj class. As for the tubes
by the instant of time, they can only be projected on static subspaces.
These projections are described in
gras.ellapx.smartdb.rels.EllUnionTubeStaticProj class. For more information
about these types of projections and their differences and for examples see
this <a class="reference internal"
href="chap_implement.html#section-label"><em>link</em></a>.</p>
+<p>These two type of objects can be projected on specified subspaces. The
projections of ellipsoid tubes can be either static or dynamic. The are
described in gras.ellapx.smartdb.rels.EllTubeProj class. As for the tubes
by instant of time, they can only be projected on static subspaces. These
projections are described in
gras.ellapx.smartdb.rels.EllUnionTubeStaticProj class. For more information
about these types of projections and their differences and for examples see
this <a class="reference internal"
href="chap_implement.html#section-label"><em>link</em></a>.</p>
<div class="section" id="ellipsoid-tubes">
<h2>Ellipsoid tubes<a class="headerlink" href="#ellipsoid-tubes"
title="Permalink to this headline">¶</a></h2>
<p>There is a whole variety of operations upon ellipsoid tubes in
<em>Ellipsoidal Toolbox</em>. We can, of course, create them. When the
ellipsoid tube object is created, we can cut them, concatenate,
interpolate, plot and project them and so on.</p>
@@ -1295,9 +1296,9 @@
</pre></div>
</td></tr></table></div>
</div>
-<div class="section" id="tubes-by-the-instant-of-time">
-<h2>Tubes by the instant of time<a class="headerlink"
href="#tubes-by-the-instant-of-time" title="Permalink to this
headline">¶</a></h2>
-<p>As with ellipsoid tube objects there are several methods that we can
use while working with tubes by the instant of time. First of all we can
create tubes by the instant of time using <em>fromEllTubes</em> method:</p>
+<div class="section" id="tubes-by-instant-of-time">
+<h2>Tubes by instant of time<a class="headerlink"
href="#tubes-by-instant-of-time" title="Permalink to this
headline">¶</a></h2>
+<p>As with ellipsoid tube objects, there are several methods that we can
use while working with tubes by instant of time. First of all we can create
tubes by instant of time using <em>fromEllTubes</em> method:</p>
<div class="highlight-matlab"><table class="highlighttable"><tr><td
class="linenos"><div class="linenodiv"><pre> 1
2
3
@@ -1327,7 +1328,7 @@
<span class="n">gras</span><span class="p">.</span><span
class="n">ellapx</span><span class="p">.</span><span
class="n">smartdb</span><span class="p">.</span><span
class="n">rels</span><span class="p">.</span><span
class="n">EllUnionTube</span><span class="p">.</span><span
class="n">fromEllTubes</span><span class="p">(</span><span
class="n">ellTubeObj</span><span class="p">);</span>
</pre></div>
</td></tr></table></div>
-<p>From here on we will use the <em>getUnion</em> function so we can get a
tube by the instant of time and work with it further on. As we have created
a tubes by the instant of time object, we can get all the types of differet
data about it. There is a set of methods that can give information about
the data stored in the object and give access to it.</p>
+<p>From here on we will use the <em>getUnion</em> function so we can get a
tube by instant of time and work with it further on. As we have created a
tubes by instant of time object, we can get all the types of differet data
about it. There is a set of methods that can give information about the
data stored in the object and give access to it.</p>
<div class="highlight-matlab"><table class="highlighttable"><tr><td
class="linenos"><div class="linenodiv"><pre>1
2
3</pre></div></td><td class="code"><div class="highlight"><pre><span
class="c">% An example of GETDATA method's usage.</span>
@@ -1601,7 +1602,7 @@
<span class="n">ellUnionObj</span><span class="p">.</span><span
class="n">display</span><span class="p">();</span>
</pre></div>
</td></tr></table></div>
-<p>Also we can compare tubes by the instant of time using <em>isEqual</em>
method.</p>
+<p>Also we can compare tubes by instant of time using <em>isEqual</em>
method.</p>
<div class="highlight-matlab"><table class="highlighttable"><tr><td
class="linenos"><div class="linenodiv"><pre>1
2
3
@@ -1615,7 +1616,7 @@
<span class="n">res</span> <span class="p">=</span> <span
class="n">firstUnionObj</span><span class="p">.</span><span
class="n">isEqual</span><span class="p">(</span><span
class="n">secondUnionObj</span><span class="p">);</span>
</pre></div>
</td></tr></table></div>
-<p>At last, as it has already been said tubes by the instant of time can
be projected only on static subspaces. It can be done in two ways.</p>
+<p>At last, as it has already been said tubes by instant of time can be
projected only on static subspaces. It can be done in two ways.</p>
<div class="highlight-matlab"><table class="highlighttable"><tr><td
class="linenos"><div class="linenodiv"><pre>1
2
3
@@ -1648,7 +1649,7 @@
<span class="n">statEllTubeProjObj</span> <span class="p">=</span> <span
class="n">unionEllTubeObj</span><span class="p">.</span><span
class="n">projectStatic</span><span class="p">(</span><span
class="n">projMatList</span><span class="p">);</span>
</pre></div>
</td></tr></table></div>
-<p>As for ellipsoid tube projections, a special function is used to create
the projection matrix for tubes by the instant of time:</p>
+<p>As for ellipsoid tube projections, a special function is used to create
the projection matrix for tubes by instant of time:</p>
<div class="highlight-matlab"><table class="highlighttable"><tr><td
class="linenos"><div class="linenodiv"><pre>1
2
3
@@ -1663,15 +1664,15 @@
</pre></div>
</td></tr></table></div>
</div>
-<div class="section" id="projections-of-ellipsoid-tubes-and-unions">
-<h2>Projections of ellipsoid tubes and unions<a class="headerlink"
href="#projections-of-ellipsoid-tubes-and-unions" title="Permalink to this
headline">¶</a></h2>
-<p>As it has already been said we can create either static or dynamic
projections for ellipsoid tubes and only static projections for tubes by
the instant of time. There are several methods in <em>Ellipsoidal
Toolbox</em> for that. Most of them has already been described:</p>
+<div class="section"
id="projections-of-ellipsoid-tubes-and-tubes-by-instant-of-time">
+<h2>Projections of ellipsoid tubes and tubes by instant of time<a
class="headerlink"
href="#projections-of-ellipsoid-tubes-and-tubes-by-instant-of-time"
title="Permalink to this headline">¶</a></h2>
+<p>As it has already been said we can create either static or dynamic
projections for ellipsoid tubes and only static projections for tubes by
instant of time. There are several methods in <em>Ellipsoidal Toolbox</em>
for that. Most of them has already been described:</p>
<ul class="simple">
<li><em>project</em>, <em>projectStatic</em> and <em>projectToOrths</em>
for ellipsoid tubes;</li>
-<li><em>project</em> and <em>projectStatic</em> for tubes by the instant
of time.</li>
+<li><em>project</em> and <em>projectStatic</em> for tubes by instant of
time.</li>
</ul>
-<p>It should be mentioned that from here on all the examples are written
for ellipsoid tube projections, but their usage is the same for the
projections of tubes by the instant of time. We wiil use <em>getProj</em>
function to create ellipsoid tube projection that we will work with.</p>
-<p>As with ellipsoid tubes and tubes by the instant of time we can get all
the types of differet data about projections. There is a set of methods
that can give information about the data stored in the object and give
access to it.</p>
+<p>It should be mentioned that from here on all the examples are written
for ellipsoid tube projections, but their usage is the same for the
projections of tubes by instant of time. We wiil use <em>getProj</em>
function to create ellipsoid tube projection that we will work with.</p>
+<p>As with ellipsoid tubes and tubes by instant of time we can get all the
types of differet data about projections. There is a set of methods that
can give information about the data stored in the object and give access to
it.</p>
<div class="highlight-matlab"><table class="highlighttable"><tr><td
class="linenos"><div class="linenodiv"><pre>1
2
3</pre></div></td><td class="code"><div class="highlight"><pre><span
class="c">% An example of GETDATA method's usage.</span>
@@ -1989,6 +1990,12 @@
</td></tr></table></div>
<p>All the three methods have several properties connected to the
properties of the image, for example: transparency, color, line width and
so on.</p>
<p>To read more about the differences of these projections <a
class="reference internal"
href="chap_implement.html#goto-label"><em>goto</em></a>. Also there you can
find some illustrations for both ellTube projections and ellUnionTube
projections.</p>
+<table class="docutils citation" frame="void" id="gag2012" rules="none">
+<colgroup><col class="label" /><col /></colgroup>
+<tbody valign="top">
+<tr><td class="label"><a class="fn-backref"
href="#id1">[GAG2012]</a></td><td>Gagarinov P.V. 2012. Computation of
Alternated Reachability Tubes for Linear Control Systems under Uncertainty.
<em>Moscow University Computational Mathematics and Cybernetics</em>, 2012,
Vol. 36, No. 4, pp. 169–177.</td></tr>
+</tbody>
+</table>
</div>
</div>

@@ -2000,10 +2007,10 @@
<div class="sphinxsidebarwrapper">
<h3><a href="main_manual.html">Table Of Contents</a></h3>
<ul>
-<li><a class="reference internal" href="#">Ellipsoid tubes, tubes by the
instant of time and their projections</a><ul>
+<li><a class="reference internal" href="#">Ellipsoid tubes, tubes by
instant of time and their projections</a><ul>
<li><a class="reference internal" href="#ellipsoid-tubes">Ellipsoid
tubes</a></li>
-<li><a class="reference internal"
href="#tubes-by-the-instant-of-time">Tubes by the instant of time</a></li>
-<li><a class="reference internal"
href="#projections-of-ellipsoid-tubes-and-unions">Projections of ellipsoid
tubes and unions</a></li>
+<li><a class="reference internal" href="#tubes-by-instant-of-time">Tubes
by instant of time</a></li>
+<li><a class="reference internal"
href="#projections-of-ellipsoid-tubes-and-tubes-by-instant-of-time">Projections
of ellipsoid tubes and tubes by instant of time</a></li>
</ul>
</li>
</ul>
@@ -2048,7 +2055,7 @@
<li class="right" >
<a href="chap_examples.html" title="Examples"
>previous</a> |</li>
- <li><a href="main_manual.html">Ellipsoidal Toolbox 2.0 beta 1
documentation</a> »</li>
+ <li><a href="main_manual.html">Ellipsoidal Toolbox 2.0.1
documentation</a> »</li>
</ul>
</div>
<div class="footer">
=======================================
--- /branches/issue_133_aatanesyan/doc/_build/html/chap_ellcalc.html Mon
Feb 24 20:40:31 2014 UTC
+++ /branches/issue_133_aatanesyan/doc/_build/html/chap_ellcalc.html Thu
Mar 6 15:09:17 2014 UTC
@@ -6,7 +6,7 @@
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />

- <title>Ellipsoidal Calculus — Ellipsoidal Toolbox 2.0 beta 1
documentation</title>
+ <title>Ellipsoidal Calculus — Ellipsoidal Toolbox 2.0.1
documentation</title>

<link rel="stylesheet" href="_static/default.css" type="text/css" />
<link rel="stylesheet" href="_static/pygments.css" type="text/css" />
@@ -14,7 +14,7 @@
<script type="text/javascript">
var DOCUMENTATION_OPTIONS = {
URL_ROOT: './',
- VERSION: '2.0 beta 1',
+ VERSION: '2.0.1',
COLLAPSE_INDEX: false,
FILE_SUFFIX: '.html',
HAS_SOURCE: true
@@ -23,7 +23,7 @@
<script type="text/javascript" src="_static/jquery.js"></script>
<script type="text/javascript" src="_static/underscore.js"></script>
<script type="text/javascript" src="_static/doctools.js"></script>
- <link rel="top" title="Ellipsoidal Toolbox 2.0 beta 1 documentation"
href="index.html" />
+ <link rel="top" title="Ellipsoidal Toolbox 2.0.1 documentation"
href="index.html" />
<link rel="next" title="Reachability" href="chap_reach.html" />
<link rel="prev" title="Introduction" href="chap_intro.html" />
</head>
@@ -40,7 +40,7 @@
<li class="right" >
<a href="chap_intro.html" title="Introduction"
accesskey="P">previous</a> |</li>
- <li><a href="main_manual.html">Ellipsoidal Toolbox 2.0 beta 1
documentation</a> »</li>
+ <li><a href="main_manual.html">Ellipsoidal Toolbox 2.0.1
documentation</a> »</li>
</ul>
</div>

@@ -1064,7 +1064,7 @@
<li class="right" >
<a href="chap_intro.html" title="Introduction"
>previous</a> |</li>
- <li><a href="main_manual.html">Ellipsoidal Toolbox 2.0 beta 1
documentation</a> »</li>
+ <li><a href="main_manual.html">Ellipsoidal Toolbox 2.0.1
documentation</a> »</li>
</ul>
</div>
<div class="footer">
=======================================
--- /branches/issue_133_aatanesyan/doc/_build/html/chap_examples.html Wed
Mar 5 22:30:52 2014 UTC
+++ /branches/issue_133_aatanesyan/doc/_build/html/chap_examples.html Thu
Mar 6 15:09:17 2014 UTC
@@ -6,7 +6,7 @@
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />

- <title>Examples — Ellipsoidal Toolbox 2.0 beta 1
documentation</title>
+ <title>Examples — Ellipsoidal Toolbox 2.0.1 documentation</title>

<link rel="stylesheet" href="_static/default.css" type="text/css" />
<link rel="stylesheet" href="_static/pygments.css" type="text/css" />
@@ -14,7 +14,7 @@
<script type="text/javascript">
var DOCUMENTATION_OPTIONS = {
URL_ROOT: './',
- VERSION: '2.0 beta 1',
+ VERSION: '2.0.1',
COLLAPSE_INDEX: false,
FILE_SUFFIX: '.html',
HAS_SOURCE: true
@@ -23,7 +23,7 @@
<script type="text/javascript" src="_static/jquery.js"></script>
<script type="text/javascript" src="_static/underscore.js"></script>
<script type="text/javascript" src="_static/doctools.js"></script>
- <link rel="top" title="Ellipsoidal Toolbox 2.0 beta 1 documentation"
href="index.html" />
+ <link rel="top" title="Ellipsoidal Toolbox 2.0.1 documentation"
href="index.html" />
<link rel="next" title="Ellipsoid tubes, tubes by the instant of time
and their projections" href="chap_ellTube.html" />
<link rel="prev" title="Implementation" href="chap_implement.html" />
</head>
@@ -40,7 +40,7 @@
<li class="right" >
<a href="chap_implement.html" title="Implementation"
accesskey="P">previous</a> |</li>
- <li><a href="main_manual.html">Ellipsoidal Toolbox 2.0 beta 1
documentation</a> »</li>
+ <li><a href="main_manual.html">Ellipsoidal Toolbox 2.0.1
documentation</a> »</li>
</ul>
</div>

@@ -72,14 +72,14 @@
<img class="math"
src="_images/math/73f5e249c88b2b3068263480f576b051cb5c4f6e.png" alt="U"/>
is the control set.</p>
<div class="figure" id="ellpolyfig">
<a class="reference internal image-reference"
href="_images/chapter06_section01_ellpoly.png"><img alt="ellpoly"
src="_images/chapter06_section01_ellpoly.png" style="width: 50%;" /></a>
-<p class="caption">Figure 1: Reach set computation performance
comparison.</p>
+<p class="caption">Figure 21: Reach set computation performance
comparison.</p>
</div>
<p>Let <img class="math"
src="_images/math/1eae27d722cf74685b3dab099d6e0089009498ce.png"
alt="{\mathcal X}_0"/> and <img class="math"
src="_images/math/73f5e249c88b2b3068263480f576b051cb5c4f6e.png" alt="U"/>
be unit boxes in
<img class="math"
src="_images/math/0798dced39715ef937fb976226ba2ae17f50367e.png" alt="{\bf
R}^2"/>, and compute the reach set using the polytope method
implemented in MPT <a class="reference internal"
href="chap_install.html#mpthp" id="id1">[MPTHP]</a>. With every
time step the number of vertices of the reach set polytope increases by
<img class="math"
src="_images/math/51a8d7e791ab9e463af82a4075edb04ef8028dcf.png" alt="4"/>.
The complexity of the convex hull computation increases
-exponentially with number of vertices. In <a href="#ellpolyfig">figure
1</a>, the time
+exponentially with number of vertices. In <a href="#ellpolyfig">figure
21</a>, the time
required to compute the reach set for different time steps using
polytopes is shown in red.</p>
<p>To compute the reach set of the system using <em>Ellipsoidal
Toolbox</em>, we
@@ -110,10 +110,10 @@
<span class="n">rsObj</span> <span class="p">=</span> <span
class="n">elltool</span><span class="p">.</span><span
class="n">reach</span><span class="p">.</span><span
class="n">ReachDiscrete</span><span class="p">(</span><span
class="n">lsys</span><span class="p">,</span> <span
class="n">x0EllObj</span><span class="p">,</span> <span
class="n">dirsMat</span><span class="p">,</span> <span
class="p">[</span><span class="mi">0</span> <span
class="n">nSteps</span><span class="p">]);</span>
</pre></div>
</td></tr></table></div>
-<p>In <a href="#ellpolyfig">figure 1</a>, the time required to compute
both external and
+<p>In <a href="#ellpolyfig">figure 21</a>, the time required to compute
both external and
internal ellipsoidal approximations, with <img class="math"
src="_images/math/3c09e376094727d14b65cedafaa5fe3a371f029d.png" alt="32"/>
ellipsoids each,
for different number of time steps is shown in blue.</p>
-<p><a href="#ellpolyfig">Figure 1</a> illustrates the fact that the
complexity of polytope
+<p><a href="#ellpolyfig">Figure 21</a> illustrates the fact that the
complexity of polytope
method grows exponentially with number of time steps, whereas the
complexity of ellipsoidal method grows linearly.</p>
</div>
@@ -121,9 +121,9 @@
<h2>System with Disturbance<a class="headerlink"
href="#system-with-disturbance" title="Permalink to this
headline">¶</a></h2>
<div class="figure" id="springmassfig">
<a class="reference internal image-reference"
href="_images/chapter06_section02_springmass.png"><img alt="spmass"
src="_images/chapter06_section02_springmass.png" style="width: 30%;" /></a>
-<p class="caption">Figure 2: Spring-mass system.</p>
+<p class="caption">Figure 22: Spring-mass system.</p>
</div>
-<p>The mechanical system presented in <a href="#springmassfig">figure
2</a>, is described
+<p>The mechanical system presented in <a href="#springmassfig">figure
22</a>, is described
by the following system of equations:</p>
<div class="math" id="equation-spmass1">
<p><span class="eqno">(1)</span><img
src="_images/math/4f3588da0ec0fdadb63756f8d4497b0620473dd2.png"
alt="m_1\ddot{x}_1+(k_1+k_2)x_1-k_2x_2 & = u_1,"/></p>
@@ -161,7 +161,7 @@
and taking its projection onto <img class="math"
src="_images/math/8718591ebf3c77219aa6da4ca77b2fba7fd6bd40.png" alt="(x_1,
x_2)"/> subspace.</p>
<div class="figure" id="mechreachfig">
<a class="reference internal image-reference"
href="_images/chapter06_section02_reachmech.png"><img alt="reachmech"
src="_images/chapter06_section02_reachmech.png" style="width: 70%;" /></a>
-<p class="caption">Figure 3: Spring-mass system without disturbance:
+<p class="caption">Figure 23: Spring-mass system without disturbance:
(a) reach tube for time <img class="math"
src="_images/math/9aed1795e3ef0707333b8278441df4a3c5071c78.png"
alt="t\in[0,4]"/>; (b) reach set at time <img class="math"
src="_images/math/c065a45befa6330020834dcf2f588a0076245e31.png" alt="t=4"/>.
Spring-mass system with disturbance:
(c) reach tube for time <img class="math"
src="_images/math/9aed1795e3ef0707333b8278441df4a3c5071c78.png"
alt="t\in[0,4]"/>; (d) reach set at time <img class="math"
src="_images/math/c065a45befa6330020834dcf2f588a0076245e31.png"
alt="t=4"/>.</p>
@@ -237,9 +237,9 @@
<span class="c">%psObj.plotByEa('g');</span>
</pre></div>
</td></tr></table></div>
-<p><a href="#mechreachfig">Figure 3</a> (a) shows the reach set of the
system
+<p><a href="#mechreachfig">Figure 23</a> (a) shows the reach set of the
system
<a href="#equation-spmass1">(1)</a>-<a href="#equation-spmass2">(2)</a>
evolving in time from <img class="math"
src="_images/math/4e2677b4edfbb85a135b7807d780a44ffa83f052.png" alt="t=0"/>
to <img class="math"
src="_images/math/c065a45befa6330020834dcf2f588a0076245e31.png" alt="t=4"/>.
-<a href="#mechreachfig">Figure 3</a> (b) presents a snapshot of this
reach set at time
+<a href="#mechreachfig">Figure 23</a> (b) presents a snapshot of this
reach set at time
<img class="math"
src="_images/math/c065a45befa6330020834dcf2f588a0076245e31.png"
alt="t=4"/>.</p>
<p>So far we considered an ideal system without any disturbance, such as
friction. We introduce disturbance to <a
href="#equation-spmass1">(1)</a>-<a href="#equation-spmass2">(2)</a> by
adding
@@ -313,19 +313,19 @@
<span class="c">%psdCutObj.plotByEa();</span>
</pre></div>
</td></tr></table></div>
-<p><a href="#mechreachfig">Figure 3</a> (c) shows the reach set of the
system
+<p><a href="#mechreachfig">Figure 23</a> (c) shows the reach set of the
system
<a href="#equation-smdist1">(4)</a>-<a href="#equation-smdist2">(5)</a>
evolving in time from <img class="math"
src="_images/math/4e2677b4edfbb85a135b7807d780a44ffa83f052.png" alt="t=0"/>
to <img class="math"
src="_images/math/c065a45befa6330020834dcf2f588a0076245e31.png" alt="t=4"/>.
-<a href="#mechreachfig">Figure 3</a> (d) presents a snapshot of this
reach set at time
+<a href="#mechreachfig">Figure 23</a> (d) presents a snapshot of this
reach set at time
<img class="math"
src="_images/math/c065a45befa6330020834dcf2f588a0076245e31.png"
alt="t=4"/>.</p>
</div>
<div class="section" id="switched-system">
<h2>Switched System<a class="headerlink" href="#switched-system"
title="Permalink to this headline">¶</a></h2>
<div class="figure" id="rlcfig">
<a class="reference internal image-reference"
href="_images/chapter06_section03_rlc.png"><img alt="rlc"
src="_images/chapter06_section03_rlc.png" style="width: 30%;" /></a>
-<p class="caption">Figure 4: RLC circuit with two inputs.</p>
+<p class="caption">Figure 24: RLC circuit with two inputs.</p>
</div>
<p>By <em>switched systems</em> we mean systems whose dynamics changes at
known
-times. Consider the RLC circuit shown in <a href="#rlcfig">figure 4</a>.
It has two
+times. Consider the RLC circuit shown in <a href="#rlcfig">figure 24</a>.
It has two
inputs - the voltage (<img class="math"
src="_images/math/0e22076955898e6c9bb38aa079135195c24dc81e.png" alt="v"/>)
and current (<img class="math"
src="_images/math/a581f053bbfa5115f42c13094857cdd12a37ec49.png" alt="i"/>)
sources. Define</p>
<ul class="simple">
<li><img class="math"
src="_images/math/b31b1ea8e4fe3f618dedfb0e2d8c69a3df7a391d.png" alt="x_1"/>
- voltage across capacitor <img class="math"
src="_images/math/a37c919e07a75380817840dfff31d2781ee02586.png"
alt="C_1"/>, so
@@ -533,17 +533,17 @@
</td></tr></table></div>
<div class="figure" id="rlcreachfig">
<a class="reference internal image-reference"
href="_images/chapter06_section03_rlcreach.png"><img alt="rlcreach"
src="_images/chapter06_section03_rlcreach.png" style="width: 80%;" /></a>
-<p class="caption">Figure 5: Forward and backward reach sets of the
switched system
+<p class="caption">Figure 25: Forward and backward reach sets of the
switched system
(external and internal approximations).</p>
</div>
-<p><a href="#rlcreachfig">Figure 5</a> (a) shows how the reach set
projection onto
+<p><a href="#rlcreachfig">Figure 25</a> (a) shows how the reach set
projection onto
<img class="math"
src="_images/math/8718591ebf3c77219aa6da4ca77b2fba7fd6bd40.png" alt="(x_1,
x_2)"/> of system <a href="#equation-rlceq2">(7)</a> evolves in time from
<img class="math"
src="_images/math/4e2677b4edfbb85a135b7807d780a44ffa83f052.png" alt="t=0"/>
to <img class="math"
src="_images/math/a95a6c7171c13a2e4d31779d50b05185acf84fd9.png"
alt="t=3"/>. The external reach set approximation for the first
dynamics is in red, the internal approximation is in green. The dynamics
switches at <img class="math"
src="_images/math/869dcc7e8055253a227de9144ac7312e31300c11.png"
alt="t=2"/>. The external reach set approximation for the
second dynamics is in yellow, its internal approximation is in blue. The
full three-dimensional external (yellow) and internal (blue)
-approximations of the reach set are shown in <a
href="#rlcreachfig">figure 5</a> (b).</p>
+approximations of the reach set are shown in <a
href="#rlcreachfig">figure 25</a> (b).</p>
<p>To find out where the system should start at time <img class="math"
src="_images/math/4e2677b4edfbb85a135b7807d780a44ffa83f052.png" alt="t=0"/>
in order
to reach a neighborhood M of the origin at time <img class="math"
src="_images/math/a95a6c7171c13a2e4d31779d50b05185acf84fd9.png"
alt="t=3"/>, we compute
the backward reach set from <img class="math"
src="_images/math/a95a6c7171c13a2e4d31779d50b05185acf84fd9.png" alt="t=3"/>
to <img class="math"
src="_images/math/4e2677b4edfbb85a135b7807d780a44ffa83f052.png"
alt="t=0"/>.</p>
@@ -658,18 +658,18 @@
<span class="c">% firstBrsObj.plotByIa('g');</span>
</pre></div>
</td></tr></table></div>
-<p><a href="#rlcreachfig">Figure 5</a> (c) presents the evolution of the
reach set projection onto
+<p><a href="#rlcreachfig">Figure 25</a> (c) presents the evolution of the
reach set projection onto
<img class="math"
src="_images/math/8718591ebf3c77219aa6da4ca77b2fba7fd6bd40.png" alt="(x_1,
x_2)"/> in backward time. Again, external and internal
approximations corresponding to the first dynamics are shown in red and
green, and to the second dynamics in yellow and blue. The full
dimensional backward reach set external and internal approximations of
-system <a href="#equation-rlceq2">(7)</a> at time <img class="math"
src="_images/math/4e2677b4edfbb85a135b7807d780a44ffa83f052.png" alt="t=0"/>
is shown in <a href="#rlcreachfig">figure 5</a> (d).</p>
+system <a href="#equation-rlceq2">(7)</a> at time <img class="math"
src="_images/math/4e2677b4edfbb85a135b7807d780a44ffa83f052.png" alt="t=0"/>
is shown in <a href="#rlcreachfig">figure 25</a> (d).</p>
</div>
<div class="section" id="hybrid-system">
<h2>Hybrid System<a class="headerlink" href="#hybrid-system"
title="Permalink to this headline">¶</a></h2>
<div class="figure" id="hwfig">
<a class="reference internal image-reference"
href="_images/chapter06_section04_hw.png"><img alt="highway"
src="_images/chapter06_section04_hw.png" style="width: 30%;" /></a>
-<p class="caption">Figure 6: Highway model. Adapted from <a
class="reference internal" href="#sun2003" id="id2">[SUN2003]</a>.</p>
+<p class="caption">Figure 26: Highway model. Adapted from <a
class="reference internal" href="#sun2003" id="id2">[SUN2003]</a>.</p>
</div>
<p>There is no explicit implementation of the reachability analysis for
hybrid systems in the <em>Ellipsoidal Toolbox</em>. Nonetheless, the
operations
@@ -679,7 +679,7 @@
polytopes.</p>
<p>We consider the <em>switching-mode model</em> of highway traffic
presented in
<a class="reference internal" href="#sun2003" id="id3">[SUN2003]</a>. The
highway segment is divided into <img class="math"
src="_images/math/75e27f04188974063be3230dca208cd495b77ce1.png" alt="N"/>
-cells as shown in <a href="#hwfig">figure 6</a>. In this particular case,
<img class="math"
src="_images/math/4a16cf007ee3e23dd0c298f6c2f9a49157a34980.png" alt="N=4"/>.
+cells as shown in <a href="#hwfig">figure 26</a>. In this particular
case, <img class="math"
src="_images/math/4a16cf007ee3e23dd0c298f6c2f9a49157a34980.png" alt="N=4"/>.
The traffic density in cell <img class="math"
src="_images/math/a581f053bbfa5115f42c13094857cdd12a37ec49.png" alt="i"/>
is <img class="math"
src="_images/math/33dfc32d00ebd5c5791c824010a155d9e5630b6f.png" alt="x_i"/>
vehicles per mile,
<img class="math"
src="_images/math/69994cf9835b884f77d639a114dd61df37c3c66c.png"
alt="i=1,2,3,4"/>.</p>
<p>Define</p>
@@ -899,7 +899,7 @@
</td></tr></table></div>
<div class="figure" id="hwreachfig">
<a class="reference internal image-reference"
href="_images/chapter06_section04_hwreach.png"><img alt="highway"
src="_images/chapter06_section04_hwreach.png" style="width: 80%;" /></a>
-<p class="caption">Figure 7: Reach set of the free-flow system is blue,
reach set of the congested
+<p class="caption">Figure 27: Reach set of the free-flow system is blue,
reach set of the congested
system is green, the guard is red.
(a) Reach set of the free-flow system at <img class="math"
src="_images/math/ddd4f149795f87346d64647fe87b53d41a0ffbc5.png" alt="t =
10"/>, before reaching the guard
(projection onto <img class="math"
src="_images/math/d244b8c7cc7aef2f4f86ee54409898356d38c63f.png"
alt="(x_1,x_2,x_3)"/>).
@@ -914,11 +914,11 @@
<p>Analyzing the values in array dVec, we conclude that the free-flow reach
set has nonempty intersection with hyperplane grdHyp at <img class="math"
src="_images/math/d7c015e171c5e9be4e19358f0d4820d6e045742a.png"
alt="t=18"/> for
the first time, and at <img class="math"
src="_images/math/6b99100f2c7c81a91f63b5e96ba1674e4cf07556.png"
alt="t=68"/> for the last time. Between
-<img class="math"
src="_images/math/d7c015e171c5e9be4e19358f0d4820d6e045742a.png"
alt="t=18"/> and <img class="math"
src="_images/math/6b99100f2c7c81a91f63b5e96ba1674e4cf07556.png"
alt="t=68"/> it crosses the guard. <a href="#hwreachfig">Figure 7</a> (a)
+<img class="math"
src="_images/math/d7c015e171c5e9be4e19358f0d4820d6e045742a.png"
alt="t=18"/> and <img class="math"
src="_images/math/6b99100f2c7c81a91f63b5e96ba1674e4cf07556.png"
alt="t=68"/> it crosses the guard. <a href="#hwreachfig">Figure 27</a> (a)
shows the free-flow reach set projection onto
<img class="math"
src="_images/math/d244b8c7cc7aef2f4f86ee54409898356d38c63f.png"
alt="(x_1,x_2,x_3)"/> subspace for <img class="math"
src="_images/math/0e8b44a3c8dcb0228b2c03537cc6e24d48740dfb.png"
alt="t=10"/>, before the guard
-crossing; <a href="#hwreachfig">figure 7</a> (b) for <img class="math"
src="_images/math/b47974de990ffde8b97693d59aa911a0a3a0c90e.png"
alt="t=50"/>, during the guard
-crossing; and <a href="#hwreachfig">figure 7</a> (c) for <img
class="math"
src="_images/math/d8131522ce7a7a504fc3982e7d0813d8a328342b.png"
alt="t=80"/>, after the guard
+crossing; <a href="#hwreachfig">figure 27</a> (b) for <img class="math"
src="_images/math/b47974de990ffde8b97693d59aa911a0a3a0c90e.png"
alt="t=50"/>, during the guard
+crossing; and <a href="#hwreachfig">figure 27</a> (c) for <img
class="math"
src="_images/math/d8131522ce7a7a504fc3982e7d0813d8a328342b.png"
alt="t=80"/>, after the guard
was crossed.</p>
<p>For each time step that the intersection of the free-flow reach set and
the guard is nonempty, we establish a new initial time and a set of
@@ -1071,7 +1071,7 @@
congested dynamics.</p>
<p>A summary of the reach set computation of the linear hybrid system
<a href="#equation-fflow">(8)</a>-<a href="#equation-cflow">(9)</a> for
<img class="math"
src="_images/math/b7935b9d18274f0d7917c7c07b9689d14205020c.png"
alt="N=100"/> time steps with one guard crossing
-is given in <a href="#hwreachfig">figure 7</a> (d), which shows the
projection of the
+is given in <a href="#hwreachfig">figure 27</a> (d), which shows the
projection of the
reach set trace onto <img class="math"
src="_images/math/0dec8dfe93d64205e7451291af8e08680e02b60b.png"
alt="(x_1,x_2)"/> subspace. The system starts
evolving in time in free-flow mode from a set of initial conditions at
<img class="math"
src="_images/math/4e2677b4edfbb85a135b7807d780a44ffa83f052.png"
alt="t=0"/>, whose boundary is shown in magenta. The free-flow reach set
@@ -1153,7 +1153,7 @@
<li class="right" >
<a href="chap_implement.html" title="Implementation"
>previous</a> |</li>
- <li><a href="main_manual.html">Ellipsoidal Toolbox 2.0 beta 1
documentation</a> »</li>
+ <li><a href="main_manual.html">Ellipsoidal Toolbox 2.0.1
documentation</a> »</li>
</ul>
</div>
<div class="footer">
=======================================
--- /branches/issue_133_aatanesyan/doc/_build/html/chap_functions.html Wed
Mar 5 22:30:52 2014 UTC
+++ /branches/issue_133_aatanesyan/doc/_build/html/chap_functions.html Thu
Mar 6 15:09:17 2014 UTC
@@ -6,7 +6,7 @@
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />

- <title>Function Reference — Ellipsoidal Toolbox 2.0 beta 1
documentation</title>
+ <title>Function Reference — Ellipsoidal Toolbox 2.0.1
documentation</title>

<link rel="stylesheet" href="_static/default.css" type="text/css" />
<link rel="stylesheet" href="_static/pygments.css" type="text/css" />
@@ -14,7 +14,7 @@
<script type="text/javascript">
var DOCUMENTATION_OPTIONS = {
URL_ROOT: './',
- VERSION: '2.0 beta 1',
+ VERSION: '2.0.1',
COLLAPSE_INDEX: false,
FILE_SUFFIX: '.html',
HAS_SOURCE: true
@@ -23,7 +23,7 @@
<script type="text/javascript" src="_static/jquery.js"></script>
<script type="text/javascript" src="_static/underscore.js"></script>
<script type="text/javascript" src="_static/doctools.js"></script>
- <link rel="top" title="Ellipsoidal Toolbox 2.0 beta 1 documentation"
href="index.html" />
+ <link rel="top" title="Ellipsoidal Toolbox 2.0.1 documentation"
href="index.html" />
<link rel="prev" title="Acknowledgement" href="chap_acknowledge.html"
/>
</head>
<body>
@@ -36,7 +36,7 @@
<li class="right" >
<a href="chap_acknowledge.html" title="Acknowledgement"
accesskey="P">previous</a> |</li>
- <li><a href="main_manual.html">Ellipsoidal Toolbox 2.0 beta 1
documentation</a> »</li>
+ <li><a href="main_manual.html">Ellipsoidal Toolbox 2.0.1
documentation</a> »</li>
</ul>
</div>

@@ -10468,7 +10468,7 @@
<li class="right" >
<a href="chap_acknowledge.html" title="Acknowledgement"
>previous</a> |</li>
- <li><a href="main_manual.html">Ellipsoidal Toolbox 2.0 beta 1
documentation</a> »</li>
+ <li><a href="main_manual.html">Ellipsoidal Toolbox 2.0.1
documentation</a> »</li>
</ul>
</div>
<div class="footer">
=======================================
--- /branches/issue_133_aatanesyan/doc/_build/html/chap_implement.html Wed
Mar 5 22:30:52 2014 UTC
+++ /branches/issue_133_aatanesyan/doc/_build/html/chap_implement.html Thu
Mar 6 15:09:17 2014 UTC
@@ -6,7 +6,7 @@
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />

- <title>Implementation — Ellipsoidal Toolbox 2.0 beta 1
documentation</title>
+ <title>Implementation — Ellipsoidal Toolbox 2.0.1
documentation</title>

<link rel="stylesheet" href="_static/default.css" type="text/css" />
<link rel="stylesheet" href="_static/pygments.css" type="text/css" />
@@ -14,7 +14,7 @@
<script type="text/javascript">
var DOCUMENTATION_OPTIONS = {
URL_ROOT: './',
- VERSION: '2.0 beta 1',
+ VERSION: '2.0.1',
COLLAPSE_INDEX: false,
FILE_SUFFIX: '.html',
HAS_SOURCE: true
@@ -23,7 +23,7 @@
<script type="text/javascript" src="_static/jquery.js"></script>
<script type="text/javascript" src="_static/underscore.js"></script>
<script type="text/javascript" src="_static/doctools.js"></script>
- <link rel="top" title="Ellipsoidal Toolbox 2.0 beta 1 documentation"
href="index.html" />
+ <link rel="top" title="Ellipsoidal Toolbox 2.0.1 documentation"
href="index.html" />
<link rel="next" title="Examples" href="chap_examples.html" />
<link rel="prev" title="Installation" href="chap_install.html" />
</head>
@@ -40,7 +40,7 @@
<li class="right" >
<a href="chap_install.html" title="Installation"
accesskey="P">previous</a> |</li>
- <li><a href="main_manual.html">Ellipsoidal Toolbox 2.0 beta 1
documentation</a> »</li>
+ <li><a href="main_manual.html">Ellipsoidal Toolbox 2.0.1
documentation</a> »</li>
</ul>
</div>

@@ -333,9 +333,9 @@
</td></tr></table></div>
<div class="figure" id="minksumpic">
<a class="reference internal image-reference"
href="_images/chapter05_section01_minksum.png"><img alt="minksum"
src="_images/chapter05_section01_minksum.png" style="width: 40%;" /></a>
-<p class="caption">Figure 8: The geometric sum of ellipsoids.</p>
+<p class="caption">Figure 35: The geometric sum of ellipsoids.</p>
</div>
-<p><a href="#minksumpic">Figure 8</a> displays the geometric sum of
ellipsoids. If
+<p><a href="#minksumpic">Figure 35</a> displays the geometric sum of
ellipsoids. If
the dimension of the space in which the ellipsoids are defined exceeds
<img class="math"
src="_images/math/b9b358d9bbdf54c3d9aef7554638822d996c21ea.png" alt="3"/>,
an error is returned. The result of the geometric sum
operation is not generally an ellipsoid, but it can be approximated by
@@ -417,9 +417,9 @@
</td></tr></table></div>
<div class="figure" id="minkdiffpic">
<a class="reference internal image-reference"
href="_images/chapter05_section01_minkdiff.png"><img alt="minkdiff"
src="_images/chapter05_section01_minkdiff.png" style="width: 40%;" /></a>
-<p class="caption">Figure 9: The geometric difference of ellipsoids.</p>
+<p class="caption">Figure 36: The geometric difference of ellipsoids.</p>
</div>
-<p><a href="#minkdiffpic">Figure 9</a> shows the geometric difference of
ellipsoids.</p>
+<p><a href="#minkdiffpic">Figure 36</a> shows the geometric difference of
ellipsoids.</p>
<p>Similar to minksum, minkdiff is there for visualization purpose. It
works only for dimensions <img class="math"
src="_images/math/123e375e88da91240024aaf085abee194c4a2d06.png" alt="1"/>,
<img class="math"
src="_images/math/15c663954a3e059d1f876bc8a4621de376038c96.png" alt="2"/>
and <img class="math"
src="_images/math/b9b358d9bbdf54c3d9aef7554638822d996c21ea.png" alt="3"/>,
and for
higher dimensions it returns an error. For arbitrary dimensions, the
@@ -505,9 +505,9 @@
</td></tr></table></div>
<div class="figure" id="minkpic">
<a class="reference internal image-reference"
href="_images/chapter05_section01_minkpmminkmp.png"><img alt="minkpmminkmp"
src="_images/chapter05_section01_minkpmminkmp.png" style="width: 100%;"
/></a>
-<p class="caption">Figure 10: Implementation of operations
‘sum-difference’ and ‘difference-sum’.</p>
+<p class="caption">Figure 37: Implementation of operations
‘sum-difference’ and ‘difference-sum’.</p>
</div>
-<p>Figure <a href="#minkpic"> 10</a> displays results of
+<p>Figure <a href="#minkpic"> 37</a> displays results of
the implementation of minkpm and minkmp operations.</p>
<p>Similarly, operation ’sum-difference’ described in section <a
class="reference internal"
href="chap_ellcalc.html#sum-diff-label"><em>Geometric
Sum-Difference</em></a> is implemented in functions
minkpm, minkpm_ea, minkpm_ia, the first one of which is used for
@@ -974,8 +974,8 @@
describe ellipsoidal tubes. The class
gras.ellapx.smartdb.rels.EllUnionTube is used to store tubes by the
instant of time:</p>
-<div class="math" id="union-label">
-<p><img src="_images/math/53ca07410d079433f63db3afd5fd5ba975724616.png"
alt="{\mathcal X}_{U}[t]=\bigcup \limits_{\tau\leqslant t}{\mathcal
X}[\tau],"/></p>
+<div class="math" id="equation-ellUnion">
+<span id="union-label"></span><p><span class="eqno">(1)</span><img
src="_images/math/53ca07410d079433f63db3afd5fd5ba975724616.png"
alt="{\mathcal X}_{U}[t]=\bigcup \limits_{\tau\leqslant t}{\mathcal
X}[\tau],"/></p>
</div><p>where <img class="math"
src="_images/math/456d10b00ec6c948f7306d194e252da674e291d1.png"
alt="{\mathcal X}[\tau]"/> is single ellipsoidal tube. The class
gras.ellapx.smartdb.rels.EllTubeProj is used to describe the projection
of the ellipsoidal tubes onto time dependent subspaces.There are two
@@ -1114,9 +1114,9 @@
tube onto time-dependent subspace.</p>
<div class="figure" id="statdyn-proj">
<a class="reference internal image-reference"
href="_images/chapter05_section03_reachTubeStatProjreachTubeDynProj.png"><img
alt="reachTubeStatProjreachTubeDynProj"
src="_images/chapter05_section03_reachTubeStatProjreachTubeDynProj.png"
style="width: 100%;" /></a>
-<p class="caption">Figure 11: Static and dynamic projections of the
ellipsoidal tube.</p>
+<p class="caption">Figure 38: Static and dynamic projections of the
ellipsoidal tube.</p>
</div>
-<p>Figure <a href="#statdyn-proj"> 11</a> displays static and dynamic
projections.
+<p>Figure <a href="#statdyn-proj"> 38</a> displays static and dynamic
projections.
Also we can see projections of good directions for ellipsoidal tubes.</p>
<p>We can compute tubes by the instant of time using
methodfromEllTubes:</p>
<div class="highlight-matlab"><table class="highlighttable"><tr><td
class="linenos"><div class="linenodiv"><pre> 1
@@ -1176,9 +1176,9 @@
</td></tr></table></div>
<div class="figure" id="uniontubestatproj">
<a class="reference internal image-reference"
href="_images/chapter05_section03_unionTubeStatProj.png"><img
alt="unionTubeStatProj"
src="_images/chapter05_section03_unionTubeStatProj.png" style="width: 70%;"
/></a>
-<p class="caption">Figure 12: Ellipsoidal tubes by the instant of time.</p>
+<p class="caption">Figure 39: Ellipsoidal tubes by the instant of time.</p>
</div>
-<p><a href="#uniontubestatproj">Figure 12</a> shows projection of
ellipsoidal
+<p><a href="#uniontubestatproj">Figure 39</a> shows projection of
ellipsoidal
tubes by the instant of time.</p>
<p>Also we can get initial data from the resulting tube:</p>
<div class="highlight-matlab"><table class="highlighttable"><tr><td
class="linenos"><div class="linenodiv"><pre>1
@@ -1227,9 +1227,9 @@
</td></tr></table></div>
<div class="figure" id="disppic">
<a class="reference internal image-reference"
href="_images/chapter05_section03_dispPic.png"><img alt="disp"
src="_images/chapter05_section03_dispPic.png" style="width: 50%;" /></a>
-<p class="caption">Figure 13: Content of the ellipsoidal tube.</p>
+<p class="caption">Figure 40: Content of the ellipsoidal tube.</p>
</div>
-<p><a href="#disppic">Figure 13</a>
+<p><a href="#disppic">Figure 40</a>
displays all fields of the ellipsoidal tube.</p>
<p>There are several methods to find the tubes with necessary
parameters.</p>
<div class="highlight-matlab"><table class="highlighttable"><tr><td
class="linenos"><div class="linenodiv"><pre> 1
@@ -2005,7 +2005,7 @@
<li class="right" >
<a href="chap_install.html" title="Installation"
>previous</a> |</li>
- <li><a href="main_manual.html">Ellipsoidal Toolbox 2.0 beta 1
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+ <li><a href="main_manual.html">Ellipsoidal Toolbox 2.0.1
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</ul>
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<div class="footer">
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--- /branches/issue_133_aatanesyan/doc/_build/html/chap_install.html Mon
Feb 24 20:40:31 2014 UTC
+++ /branches/issue_133_aatanesyan/doc/_build/html/chap_install.html Thu
Mar 6 15:09:17 2014 UTC
@@ -6,7 +6,7 @@
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />

- <title>Installation — Ellipsoidal Toolbox 2.0 beta 1
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+ <title>Installation — Ellipsoidal Toolbox 2.0.1
documentation</title>

<link rel="stylesheet" href="_static/default.css" type="text/css" />
<link rel="stylesheet" href="_static/pygments.css" type="text/css" />
@@ -14,7 +14,7 @@
<script type="text/javascript">
var DOCUMENTATION_OPTIONS = {
URL_ROOT: './',
- VERSION: '2.0 beta 1',
+ VERSION: '2.0.1',
COLLAPSE_INDEX: false,
FILE_SUFFIX: '.html',
HAS_SOURCE: true
@@ -23,7 +23,7 @@
<script type="text/javascript" src="_static/jquery.js"></script>
<script type="text/javascript" src="_static/underscore.js"></script>
<script type="text/javascript" src="_static/doctools.js"></script>
- <link rel="top" title="Ellipsoidal Toolbox 2.0 beta 1 documentation"
href="index.html" />
+ <link rel="top" title="Ellipsoidal Toolbox 2.0.1 documentation"
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<link rel="prev" title="Reachability" href="chap_reach.html" />
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@@ -40,7 +40,7 @@
<li class="right" >
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accesskey="P">previous</a> |</li>
- <li><a href="main_manual.html">Ellipsoidal Toolbox 2.0 beta 1
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+ <li><a href="main_manual.html">Ellipsoidal Toolbox 2.0.1
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</ul>
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@@ -226,7 +226,7 @@
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>previous</a> |</li>
- <li><a href="main_manual.html">Ellipsoidal Toolbox 2.0 beta 1
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<div class="footer">
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--- /branches/issue_133_aatanesyan/doc/_build/html/chap_intro.html Mon Feb
24 20:40:31 2014 UTC
+++ /branches/issue_133_aatanesyan/doc/_build/html/chap_intro.html Thu Mar
6 15:09:17 2014 UTC
@@ -6,7 +6,7 @@
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />

- <title>Introduction — Ellipsoidal Toolbox 2.0 beta 1
documentation</title>
+ <title>Introduction — Ellipsoidal Toolbox 2.0.1
documentation</title>

<link rel="stylesheet" href="_static/default.css" type="text/css" />
<link rel="stylesheet" href="_static/pygments.css" type="text/css" />
@@ -14,7 +14,7 @@
<script type="text/javascript">
var DOCUMENTATION_OPTIONS = {
URL_ROOT: './',
- VERSION: '2.0 beta 1',
+ VERSION: '2.0.1',
COLLAPSE_INDEX: false,
FILE_SUFFIX: '.html',
HAS_SOURCE: true
@@ -23,7 +23,7 @@
<script type="text/javascript" src="_static/jquery.js"></script>
<script type="text/javascript" src="_static/underscore.js"></script>
<script type="text/javascript" src="_static/doctools.js"></script>
- <link rel="top" title="Ellipsoidal Toolbox 2.0 beta 1 documentation"
href="index.html" />
+ <link rel="top" title="Ellipsoidal Toolbox 2.0.1 documentation"
href="index.html" />
<link rel="next" title="Ellipsoidal Calculus" href="chap_ellcalc.html"
/>
<link rel="prev" title="Welcome to Ellipsoidal Toolbox documentation!"
href="main_manual.html" />
</head>
@@ -40,7 +40,7 @@
<li class="right" >
<a href="main_manual.html" title="Welcome to Ellipsoidal Toolbox
documentation!"
accesskey="P">previous</a> |</li>
- <li><a href="main_manual.html">Ellipsoidal Toolbox 2.0 beta 1
documentation</a> »</li>
+ <li><a href="main_manual.html">Ellipsoidal Toolbox 2.0.1
documentation</a> »</li>
</ul>
</div>

@@ -97,21 +97,21 @@
unions of rectangular polytopes <a class="reference internal"
href="#asar2000" id="id6">[ASAR2000]</a>.</p>
<div class="figure align-center" id="ddtfig" style="width: 50%">
<img alt="approximation" src="_images/chapter01_ddt.png" />
-<p class="caption">Figure 14: Reach set approximation by union of
rectangles. Source: adapted from <a class="reference internal"
href="#asar2000" id="id7">[ASAR2000]</a>.</p>
+<p class="caption">Figure 34: Reach set approximation by union of
rectangles. Source: adapted from <a class="reference internal"
href="#asar2000" id="id7">[ASAR2000]</a>.</p>
</div>
<p>The algorithm works as follows. First, given the set of initial
conditions defined as a polytope, the evolution in time of the
-polytope’s extreme points is computed (<a href="#ddtfig">figure 14</a>
(a)).</p>
-<p><img class="math"
src="_images/math/088e22e395ff0b8f6cde149172e52590852bcebf.png"
alt="R(t_1)"/> in <a href="#ddtfig">figure 14</a> (a) is the reach set of
the system at
+polytope’s extreme points is computed (<a href="#ddtfig">figure 34</a>
(a)).</p>
+<p><img class="math"
src="_images/math/088e22e395ff0b8f6cde149172e52590852bcebf.png"
alt="R(t_1)"/> in <a href="#ddtfig">figure 34</a> (a) is the reach set of
the system at
time <img class="math"
src="_images/math/44780f0598b4d22c6888cfca27deb04115f7fa88.png"
alt="t_1"/>, and <img class="math"
src="_images/math/1a3e51473c097540a265ee70575abb8038230e74.png" alt="R[t_0,
t_1]"/> is the set of all points that
can be reached during <img class="math"
src="_images/math/85674382bfceecdd6e6d6bdfbc6cff246ef47ef9.png" alt="[t_0,
t_1]"/>. Second, the algorithm computes
the convex hull of vertices of both, the initial polytope and
-<img class="math"
src="_images/math/088e22e395ff0b8f6cde149172e52590852bcebf.png"
alt="R(t_1)"/> (<a href="#ddtfig">figure 14</a> (b)). The resulting
polytope is then
-bloated to include all the reachable states in <img class="math"
src="_images/math/3ad5575eeecc18f7013d14bb6ac8e07d4a611af3.png"
alt="[t_0,t_1]"/> (<a href="#ddtfig">figure 14</a> (c)).
+<img class="math"
src="_images/math/088e22e395ff0b8f6cde149172e52590852bcebf.png"
alt="R(t_1)"/> (<a href="#ddtfig">figure 34</a> (b)). The resulting
polytope is then
+bloated to include all the reachable states in <img class="math"
src="_images/math/3ad5575eeecc18f7013d14bb6ac8e07d4a611af3.png"
alt="[t_0,t_1]"/> (<a href="#ddtfig">figure 34</a> (c)).
Finally, this overapproximating polytope is in its turn
-overapproximated by the union of rectangles (<a href="#ddtfig">figure
14</a> (d)). The
+overapproximated by the union of rectangles (<a href="#ddtfig">figure
34</a> (d)). The
same procedure is repeated for the next time interval <img class="math"
src="_images/math/d82d790d123640183a6478ccd2fca6d3dbbb5cc0.png"
alt="[t_1,t_2]"/>,
-and the union of both rectangular approximations is taken (<a
href="#ddtfig">figure 14</a> (e,f)),
+and the union of both rectangular approximations is taken (<a
href="#ddtfig">figure 34</a> (e,f)),
and so on. Rectangular polytopes are easy to represent
and the number of facets grows linearly with dimension, but a large
number of rectangles must be used to assure the approximation is not
@@ -438,7 +438,7 @@
<li class="right" >
<a href="main_manual.html" title="Welcome to Ellipsoidal Toolbox
documentation!"
>previous</a> |</li>
- <li><a href="main_manual.html">Ellipsoidal Toolbox 2.0 beta 1
documentation</a> »</li>
+ <li><a href="main_manual.html">Ellipsoidal Toolbox 2.0.1
documentation</a> »</li>
</ul>
</div>
<div class="footer">
=======================================
--- /branches/issue_133_aatanesyan/doc/_build/html/chap_reach.html Mon Feb
24 20:40:31 2014 UTC
+++ /branches/issue_133_aatanesyan/doc/_build/html/chap_reach.html Thu Mar
6 15:09:17 2014 UTC
@@ -6,7 +6,7 @@
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />

- <title>Reachability — Ellipsoidal Toolbox 2.0 beta 1
documentation</title>
+ <title>Reachability — Ellipsoidal Toolbox 2.0.1
documentation</title>

<link rel="stylesheet" href="_static/default.css" type="text/css" />
<link rel="stylesheet" href="_static/pygments.css" type="text/css" />
@@ -14,7 +14,7 @@
<script type="text/javascript">
var DOCUMENTATION_OPTIONS = {
URL_ROOT: './',
- VERSION: '2.0 beta 1',
+ VERSION: '2.0.1',
COLLAPSE_INDEX: false,
FILE_SUFFIX: '.html',
HAS_SOURCE: true
@@ -23,7 +23,7 @@
<script type="text/javascript" src="_static/jquery.js"></script>
<script type="text/javascript" src="_static/underscore.js"></script>
<script type="text/javascript" src="_static/doctools.js"></script>
- <link rel="top" title="Ellipsoidal Toolbox 2.0 beta 1 documentation"
href="index.html" />
+ <link rel="top" title="Ellipsoidal Toolbox 2.0.1 documentation"
href="index.html" />
<link rel="next" title="Installation" href="chap_install.html" />
<link rel="prev" title="Ellipsoidal Calculus" href="chap_ellcalc.html"
/>
</head>
@@ -40,7 +40,7 @@
<li class="right" >
<a href="chap_ellcalc.html" title="Ellipsoidal Calculus"
accesskey="P">previous</a> |</li>
- <li><a href="main_manual.html">Ellipsoidal Toolbox 2.0 beta 1
documentation</a> »</li>
+ <li><a href="main_manual.html">Ellipsoidal Toolbox 2.0.1
documentation</a> »</li>
</ul>
</div>

@@ -1414,7 +1414,7 @@
<li class="right" >
<a href="chap_ellcalc.html" title="Ellipsoidal Calculus"
>previous</a> |</li>
- <li><a href="main_manual.html">Ellipsoidal Toolbox 2.0 beta 1
documentation</a> »</li>
+ <li><a href="main_manual.html">Ellipsoidal Toolbox 2.0.1
documentation</a> »</li>
</ul>
</div>
<div class="footer">
=======================================
--- /branches/issue_133_aatanesyan/doc/_build/html/chap_summary.html Wed
Mar 5 22:30:52 2014 UTC
+++ /branches/issue_133_aatanesyan/doc/_build/html/chap_summary.html Thu
Mar 6 15:09:17 2014 UTC
@@ -6,7 +6,7 @@
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />

- <title>Summary and Outlook — Ellipsoidal Toolbox 2.0 beta 1
documentation</title>
+ <title>Summary and Outlook — Ellipsoidal Toolbox 2.0.1
documentation</title>

<link rel="stylesheet" href="_static/default.css" type="text/css" />
<link rel="stylesheet" href="_static/pygments.css" type="text/css" />
@@ -14,7 +14,7 @@
<script type="text/javascript">
var DOCUMENTATION_OPTIONS = {
URL_ROOT: './',
- VERSION: '2.0 beta 1',
+ VERSION: '2.0.1',
COLLAPSE_INDEX: false,
FILE_SUFFIX: '.html',
HAS_SOURCE: true
@@ -23,7 +23,7 @@
<script type="text/javascript" src="_static/jquery.js"></script>
<script type="text/javascript" src="_static/underscore.js"></script>
<script type="text/javascript" src="_static/doctools.js"></script>
- <link rel="top" title="Ellipsoidal Toolbox 2.0 beta 1 documentation"
href="index.html" />
+ <link rel="top" title="Ellipsoidal Toolbox 2.0.1 documentation"
href="index.html" />
<link rel="next" title="Acknowledgement" href="chap_acknowledge.html"
/>
<link rel="prev" title="Ellipsoid tubes, tubes by the instant of time
and their projections" href="chap_ellTube.html" />
</head>
@@ -40,7 +40,7 @@
<li class="right" >
<a href="chap_ellTube.html" title="Ellipsoid tubes, tubes by the
instant of time and their projections"
accesskey="P">previous</a> |</li>
- <li><a href="main_manual.html">Ellipsoidal Toolbox 2.0 beta 1
documentation</a> »</li>
+ <li><a href="main_manual.html">Ellipsoidal Toolbox 2.0.1
documentation</a> »</li>
</ul>
</div>

@@ -126,7 +126,7 @@
<li class="right" >
<a href="chap_ellTube.html" title="Ellipsoid tubes, tubes by the
instant of time and their projections"
>previous</a> |</li>
- <li><a href="main_manual.html">Ellipsoidal Toolbox 2.0 beta 1
documentation</a> »</li>
+ <li><a href="main_manual.html">Ellipsoidal Toolbox 2.0.1
documentation</a> »</li>
</ul>
</div>
<div class="footer">
=======================================
--- /branches/issue_133_aatanesyan/doc/_build/html/genindex.html Mon Feb 24
20:40:31 2014 UTC
+++ /branches/issue_133_aatanesyan/doc/_build/html/genindex.html Thu Mar 6
15:09:17 2014 UTC
@@ -7,7 +7,7 @@
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />

- <title>Index — Ellipsoidal Toolbox 2.0 beta 1
documentation</title>
+ <title>Index — Ellipsoidal Toolbox 2.0.1 documentation</title>

<link rel="stylesheet" href="_static/default.css" type="text/css" />
<link rel="stylesheet" href="_static/pygments.css" type="text/css" />
@@ -15,7 +15,7 @@
<script type="text/javascript">
var DOCUMENTATION_OPTIONS = {
URL_ROOT: './',
- VERSION: '2.0 beta 1',
+ VERSION: '2.0.1',
COLLAPSE_INDEX: false,
FILE_SUFFIX: '.html',
HAS_SOURCE: true
@@ -24,7 +24,7 @@
<script type="text/javascript" src="_static/jquery.js"></script>
<script type="text/javascript" src="_static/underscore.js"></script>
<script type="text/javascript" src="_static/doctools.js"></script>
- <link rel="top" title="Ellipsoidal Toolbox 2.0 beta 1 documentation"
href="index.html" />
+ <link rel="top" title="Ellipsoidal Toolbox 2.0.1 documentation"
href="index.html" />
</head>
<body>
<div class="related">
@@ -33,7 +33,7 @@
<li class="right" style="margin-right: 10px">
<a href="#" title="General Index"
accesskey="I">index</a></li>
- <li><a href="main_manual.html">Ellipsoidal Toolbox 2.0 beta 1
documentation</a> »</li>
+ <li><a href="main_manual.html">Ellipsoidal Toolbox 2.0.1
documentation</a> »</li>
</ul>
</div>

@@ -81,7 +81,7 @@
<li class="right" style="margin-right: 10px">
<a href="#" title="General Index"
>index</a></li>
- <li><a href="main_manual.html">Ellipsoidal Toolbox 2.0 beta 1
documentation</a> »</li>
+ <li><a href="main_manual.html">Ellipsoidal Toolbox 2.0.1
documentation</a> »</li>
</ul>
</div>
<div class="footer">
=======================================
--- /branches/issue_133_aatanesyan/doc/_build/html/main_manual.html Wed
Mar 5 22:30:52 2014 UTC
+++ /branches/issue_133_aatanesyan/doc/_build/html/main_manual.html Thu
Mar 6 15:09:17 2014 UTC
@@ -6,7 +6,7 @@
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />

- <title>Welcome to Ellipsoidal Toolbox documentation! —
Ellipsoidal Toolbox 2.0 beta 1 documentation</title>
+ <title>Welcome to Ellipsoidal Toolbox documentation! —
Ellipsoidal Toolbox 2.0.1 documentation</title>

<link rel="stylesheet" href="_static/default.css" type="text/css" />
<link rel="stylesheet" href="_static/pygments.css" type="text/css" />
@@ -14,7 +14,7 @@
<script type="text/javascript">
var DOCUMENTATION_OPTIONS = {
URL_ROOT: './',
- VERSION: '2.0 beta 1',
+ VERSION: '2.0.1',
COLLAPSE_INDEX: false,
FILE_SUFFIX: '.html',
HAS_SOURCE: true
@@ -23,7 +23,7 @@
<script type="text/javascript" src="_static/jquery.js"></script>
<script type="text/javascript" src="_static/underscore.js"></script>
<script type="text/javascript" src="_static/doctools.js"></script>
- <link rel="top" title="Ellipsoidal Toolbox 2.0 beta 1 documentation"
href="index.html" />
+ <link rel="top" title="Ellipsoidal Toolbox 2.0.1 documentation"
href="index.html" />
<link rel="next" title="Introduction" href="chap_intro.html" />
</head>
<body>
@@ -36,7 +36,7 @@
<li class="right" >
<a href="chap_intro.html" title="Introduction"
accesskey="N">next</a> |</li>
- <li><a href="#">Ellipsoidal Toolbox 2.0 beta 1 documentation</a>
»</li>
+ <li><a href="#">Ellipsoidal Toolbox 2.0.1 documentation</a>
»</li>
</ul>
</div>

@@ -107,10 +107,10 @@
<li class="toctree-l2"><a class="reference internal"
href="chap_examples.html#hybrid-system">Hybrid System</a></li>
</ul>
</li>
-<li class="toctree-l1"><a class="reference internal"
href="chap_ellTube.html">Ellipsoid tubes, tubes by the instant of time and
their projections</a><ul>
+<li class="toctree-l1"><a class="reference internal"
href="chap_ellTube.html">Ellipsoid tubes, tubes by instant of time and
their projections</a><ul>
<li class="toctree-l2"><a class="reference internal"
href="chap_ellTube.html#ellipsoid-tubes">Ellipsoid tubes</a></li>
-<li class="toctree-l2"><a class="reference internal"
href="chap_ellTube.html#tubes-by-the-instant-of-time">Tubes by the instant
of time</a></li>
-<li class="toctree-l2"><a class="reference internal"
href="chap_ellTube.html#projections-of-ellipsoid-tubes-and-unions">Projections
of ellipsoid tubes and unions</a></li>
+<li class="toctree-l2"><a class="reference internal"
href="chap_ellTube.html#tubes-by-instant-of-time">Tubes by instant of
time</a></li>
+<li class="toctree-l2"><a class="reference internal"
href="chap_ellTube.html#projections-of-ellipsoid-tubes-and-tubes-by-instant-of-time">Projections
of ellipsoid tubes and tubes by instant of time</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal"
href="chap_summary.html">Summary and Outlook</a></li>
@@ -179,7 +179,7 @@
<li class="right" >
<a href="chap_intro.html" title="Introduction"
>next</a> |</li>
- <li><a href="#">Ellipsoidal Toolbox 2.0 beta 1 documentation</a>
»</li>
+ <li><a href="#">Ellipsoidal Toolbox 2.0.1 documentation</a>
»</li>
</ul>
</div>
<div class="footer">
=======================================
--- /branches/issue_133_aatanesyan/doc/_build/html/objects.inv Wed Mar 5
22:30:52 2014 UTC
+++ /branches/issue_133_aatanesyan/doc/_build/html/objects.inv Thu Mar 6
15:09:17 2014 UTC
Binary file, no diff available.
=======================================
--- /branches/issue_133_aatanesyan/doc/_build/html/search.html Mon Feb 24
20:40:31 2014 UTC
+++ /branches/issue_133_aatanesyan/doc/_build/html/search.html Thu Mar 6
15:09:17 2014 UTC
@@ -6,7 +6,7 @@
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />

- <title>Search — Ellipsoidal Toolbox 2.0 beta 1
documentation</title>
+ <title>Search — Ellipsoidal Toolbox 2.0.1 documentation</title>

<link rel="stylesheet" href="_static/default.css" type="text/css" />
<link rel="stylesheet" href="_static/pygments.css" type="text/css" />
@@ -14,7 +14,7 @@
<script type="text/javascript">
var DOCUMENTATION_OPTIONS = {
URL_ROOT: './',
- VERSION: '2.0 beta 1',
+ VERSION: '2.0.1',
COLLAPSE_INDEX: false,
FILE_SUFFIX: '.html',
HAS_SOURCE: true
@@ -24,7 +24,7 @@
<script type="text/javascript" src="_static/underscore.js"></script>
<script type="text/javascript" src="_static/doctools.js"></script>
<script type="text/javascript" src="_static/searchtools.js"></script>
- <link rel="top" title="Ellipsoidal Toolbox 2.0 beta 1 documentation"
href="index.html" />
+ <link rel="top" title="Ellipsoidal Toolbox 2.0.1 documentation"
href="index.html" />
<script type="text/javascript">
jQuery(function() { Search.loadIndex("searchindex.js"); });
</script>
@@ -40,7 +40,7 @@
<li class="right" style="margin-right: 10px">
<a href="genindex.html" title="General Index"
accesskey="I">index</a></li>
- <li><a href="main_manual.html">Ellipsoidal Toolbox 2.0 beta 1
documentation</a> »</li>
+ <li><a href="main_manual.html">Ellipsoidal Toolbox 2.0.1
documentation</a> »</li>
</ul>
</div>

@@ -88,7 +88,7 @@
<li class="right" style="margin-right: 10px">
<a href="genindex.html" title="General Index"
>index</a></li>
- <li><a href="main_manual.html">Ellipsoidal Toolbox 2.0 beta 1
documentation</a> »</li>
+ <li><a href="main_manual.html">Ellipsoidal Toolbox 2.0.1
documentation</a> »</li>
</ul>
</div>
<div class="footer">
=======================================
--- /branches/issue_133_aatanesyan/doc/_build/html/searchindex.js Wed Mar
5 22:30:52 2014 UTC
+++ /branches/issue_133_aatanesyan/doc/_build/html/searchindex.js Thu Mar
6 15:09:17 2014 UTC
@@ -1,1 +1,1 @@
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and
Outlook","Ellipsoid tubes, tubes by instant of time and their
projections","Ellipsoidal
Calculus","Acknowledgement","Installation","Reachability","Function
Reference","Introduction","Welcome to Ellipsoidal Toolbox
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=======================================
--- /branches/issue_133_aatanesyan/doc/_build/latex/elltool_manual.tex Wed
Mar 5 22:30:52 2014 UTC
+++ /branches/issue_133_aatanesyan/doc/_build/latex/elltool_manual.tex Thu
Mar 6 15:09:17 2014 UTC
@@ -18,7 +18,7 @@

\title{Ellipsoidal Toolbox}
\date{March 06, 2014}
-\release{2.0 beta 1}
+\release{2.0.1}
\author{Peter Gagarinov, Alex A. Kurzhanskiy}
\newcommand{\sphinxlogo}{}
\renewcommand{\releasename}{Release}
@@ -187,7 +187,7 @@
\capstart

\includegraphics{chapter01_ddt.png}
-\caption{Figure 14: Reach set approximation by union of rectangles.
Source: adapted from
{\hyperref[chap_intro:asar2000]{{[}ASAR2000{]}}}.}\label{chap_intro:ddtfig}\end{figure}
+\caption{Figure 34: Reach set approximation by union of rectangles.
Source: adapted from
{\hyperref[chap_intro:asar2000]{{[}ASAR2000{]}}}.}\label{chap_intro:ddtfig}\end{figure}

The algorithm works as follows. First, given the set of initial
conditions defined as a polytope, the evolution in time of the
@@ -3283,7 +3283,7 @@
\capstart

\includegraphics[width=0.400\linewidth]{chapter05_section01_minksum.png}
-\caption{Figure 8: The geometric sum of
ellipsoids.}\label{chap_implement:minksumpic}\end{figure}
+\caption{Figure 35: The geometric sum of
ellipsoids.}\label{chap_implement:minksumpic}\end{figure}

\hyperref[chap_implement:minksumpic]{Figure
\ref*{chap_implement:minksumpic}} displays the geometric sum of ellipsoids.
If
the dimension of the space in which the ellipsoids are defined exceeds
@@ -3343,7 +3343,7 @@
\capstart

\includegraphics[width=0.400\linewidth]{chapter05_section01_minkdiff.png}
-\caption{Figure 9: The geometric difference of
ellipsoids.}\label{chap_implement:minkdiffpic}\end{figure}
+\caption{Figure 36: The geometric difference of
ellipsoids.}\label{chap_implement:minkdiffpic}\end{figure}

\hyperref[chap_implement:minkdiffpic]{Figure
\ref*{chap_implement:minkdiffpic}} shows the geometric difference of
ellipsoids.

@@ -3405,7 +3405,7 @@
\capstart

\includegraphics[width=1.000\linewidth]{chapter05_section01_minkpmminkmp.png}
-\caption{Figure 10: Implementation of operations `sum-difference' and
`difference-sum'.}\label{chap_implement:minkpic}\end{figure}
+\caption{Figure 37: Implementation of operations `sum-difference' and
`difference-sum'.}\label{chap_implement:minkpic}\end{figure}

Figure \hyperref[chap_implement:minkpic]{ \ref*{chap_implement:minkpic}}
displays results of
the implementation of minkpm and minkmp operations.
@@ -3743,8 +3743,8 @@
describe ellipsoidal tubes. The class
gras.ellapx.smartdb.rels.EllUnionTube is used to store tubes by the
instant of time:
-\phantomsection\label{chap_implement:union-label}\begin{gather}
-\begin{split}{\mathcal X}_{U}[t]=\bigcup \limits_{\tau\leqslant
t}{\mathcal X}[\tau],\end{split}\notag
+\phantomsection\label{chap_implement:union-label}\phantomsection\label{chap_implement:equation-ellUnion}\begin{gather}
+\begin{split}{\mathcal X}_{U}[t]=\bigcup \limits_{\tau\leqslant
t}{\mathcal X}[\tau],\end{split}\label{chap_implement-ellUnion}
\end{gather}
where \({\mathcal X}[\tau]\) is single ellipsoidal tube. The class
gras.ellapx.smartdb.rels.EllTubeProj is used to describe the projection
@@ -3838,7 +3838,7 @@
\capstart

\includegraphics[width=1.000\linewidth]{chapter05_section03_reachTubeStatProjreachTubeDynProj.png}
-\caption{Figure 11: Static and dynamic projections of the ellipsoidal
tube.}\label{chap_implement:statdyn-proj}\end{figure}
+\caption{Figure 38: Static and dynamic projections of the ellipsoidal
tube.}\label{chap_implement:statdyn-proj}\end{figure}

Figure \hyperref[chap_implement:statdyn-proj]{
\ref*{chap_implement:statdyn-proj}} displays static and dynamic projections.
Also we can see projections of good directions for ellipsoidal tubes.
@@ -3879,7 +3879,7 @@
\capstart

\includegraphics[width=0.700\linewidth]{chapter05_section03_unionTubeStatProj.png}
-\caption{Figure 12: Ellipsoidal tubes by the instant of
time.}\label{chap_implement:uniontubestatproj}\end{figure}
+\caption{Figure 39: Ellipsoidal tubes by the instant of
time.}\label{chap_implement:uniontubestatproj}\end{figure}

\hyperref[chap_implement:uniontubestatproj]{Figure
\ref*{chap_implement:uniontubestatproj}} shows projection of ellipsoidal
tubes by the instant of time.
@@ -3919,7 +3919,7 @@
\capstart

\includegraphics[width=0.500\linewidth]{chapter05_section03_dispPic.png}
-\caption{Figure 13: Content of the ellipsoidal
tube.}\label{chap_implement:disppic}\end{figure}
+\caption{Figure 40: Content of the ellipsoidal
tube.}\label{chap_implement:disppic}\end{figure}

\hyperref[chap_implement:disppic]{Figure \ref*{chap_implement:disppic}}
displays all fields of the ellipsoidal tube.
@@ -4562,7 +4562,7 @@
\capstart

\includegraphics[width=0.500\linewidth]{chapter06_section01_ellpoly.png}
-\caption{Figure 1: Reach set computation performance
comparison.}\label{chap_examples:ellpolyfig}\end{figure}
+\caption{Figure 21: Reach set computation performance
comparison.}\label{chap_examples:ellpolyfig}\end{figure}

Let \({\mathcal X}_0\) and \(U\) be unit boxes in
\({\bf R}^2\), and compute the reach set using the polytope method
@@ -4608,7 +4608,7 @@
\capstart

\includegraphics[width=0.300\linewidth]{chapter06_section02_springmass.png}
-\caption{Figure 2: Spring-mass
system.}\label{chap_examples:springmassfig}\end{figure}
+\caption{Figure 22: Spring-mass
system.}\label{chap_examples:springmassfig}\end{figure}

The mechanical system presented in
\hyperref[chap_examples:springmassfig]{figure
\ref*{chap_examples:springmassfig}}, is described
by the following system of equations:
@@ -4653,7 +4653,7 @@
\capstart

\includegraphics[width=0.700\linewidth]{chapter06_section02_reachmech.png}
-\caption{Figure 3: Spring-mass system without disturbance:
+\caption{Figure 23: Spring-mass system without disturbance:
(a) reach tube for time \(t\in[0,4]\); (b) reach set at time \(t=4\).
Spring-mass system with disturbance:
(c) reach tube for time \(t\in[0,4]\); (d) reach set at time
\(t=4\).}\label{chap_examples:mechreachfig}\end{figure}
@@ -4764,7 +4764,7 @@
\capstart

\includegraphics[width=0.300\linewidth]{chapter06_section03_rlc.png}
-\caption{Figure 4: RLC circuit with two
inputs.}\label{chap_examples:rlcfig}\end{figure}
+\caption{Figure 24: RLC circuit with two
inputs.}\label{chap_examples:rlcfig}\end{figure}

By \emph{switched systems} we mean systems whose dynamics changes at known
times. Consider the RLC circuit shown in
\hyperref[chap_examples:rlcfig]{figure \ref*{chap_examples:rlcfig}}. It
has two
@@ -4921,7 +4921,7 @@
\capstart

\includegraphics[width=0.800\linewidth]{chapter06_section03_rlcreach.png}
-\caption{Figure 5: Forward and backward reach sets of the switched system
+\caption{Figure 25: Forward and backward reach sets of the switched system
(external and internal
approximations).}\label{chap_examples:rlcreachfig}\end{figure}

\hyperref[chap_examples:rlcreachfig]{Figure
\ref*{chap_examples:rlcreachfig}} (a) shows how the reach set projection
onto
@@ -5009,7 +5009,7 @@
\capstart

\includegraphics[width=0.300\linewidth]{chapter06_section04_hw.png}
-\caption{Figure 6: Highway model. Adapted from
{\hyperref[chap_examples:sun2003]{{[}SUN2003{]}}}.}\label{chap_examples:hwfig}\end{figure}
+\caption{Figure 26: Highway model. Adapted from
{\hyperref[chap_examples:sun2003]{{[}SUN2003{]}}}.}\label{chap_examples:hwfig}\end{figure}

There is no explicit implementation of the reachability analysis for
hybrid systems in the \emph{Ellipsoidal Toolbox}. Nonetheless, the
operations
@@ -5214,7 +5214,7 @@
\capstart

\includegraphics[width=0.800\linewidth]{chapter06_section04_hwreach.png}
-\caption{Figure 7: Reach set of the free-flow system is blue, reach set of
the congested
+\caption{Figure 27: Reach set of the free-flow system is blue, reach set
of the congested
system is green, the guard is red.
(a) Reach set of the free-flow system at \(t = 10\), before reaching the
guard
(projection onto \((x_1,x_2,x_3)\)).
@@ -5337,25 +5337,27 @@
to the green region.

-\chapter{Ellipsoid tubes, tubes by the instant of time and their
projections}
-\label{chap_ellTube:ellipsoid-tubes-tubes-by-the-instant-of-time-and-their-projections}\label{chap_ellTube::doc}
-\textbf{Definition.} For any matrix asimptotic monotone function
\(M(\cdot)\), which defines the configuration of the environment of any
point of time, the quadratically regularized alternated reach set of system
with disturbance is
+\chapter{Ellipsoid tubes, tubes by instant of time and their projections}
+\label{chap_ellTube::doc}\label{chap_ellTube:ellipsoid-tubes-tubes-by-instant-of-time-and-their-projections}
+\textbf{Definition.} For any regularization matrix \(M(\cdot)\) the
quadratically regularized alternated reach set of system with uncertainty is
\begin{gather}
\begin{split}{\mathcal X}^{q}_{U}(t,t_0,{\mathcal X}^{0},M(\cdot)) =
\underset{t_0\leq\tau\leq t}{\bigcup}{\mathcal
X}^{q}_{U}(\tau,t_0,{\mathcal X}^{0},M (\cdot)).\end{split}\notag
\end{gather}
-Identify as \({\mathcal E}(\overline{x}(t),X_{+}(t,l))\) и \({\mathcal
E}(\overline{x}(t),X_{-}(t,l))\) the tight external and internal
approximations along \(l(\cdot)\) good direction such as \(l(t_0)=l\). Then
the reach set by instant of time can be described as
+Note that if system doesn't have uncertainty then \(M(\cdot)=0\).
+
+By \({\mathcal E}(\overline{x}(t),X_{+}(t,l))\) and \({\mathcal
E}(\overline{x}(t),X_{-}(t,l))\) denote tight external and internal
approximations along \(l(\cdot)\) good direction such as \(l(t_0)=l\). Then
the reach set by instant of time can be described as
\begin{gather}
\begin{split}{\mathcal
X}^{q}_{U}[t]=\underset{\tau}{\bigcup}\underset{l}{\bigcap}\{
{\mathcal E}(\overline{x}(\tau),X_{+}(\tau,l)) |l\in {\mathcal S}_1(0),
\tau\leq t\}\subseteq\underset{l} {\bigcap}\{{\mathcal E}^{U}_{+}[t,l]) |
l\in {\mathcal S}_1(0)\},\end{split}\notag
\end{gather}
-where \({\mathcal E}^{U}_{+}[t,l]=\underset{\tau}{\bigcup}\{{\mathcal
E}(\overline{x}(\tau),X_{+}(\tau,l))|t_0\leq\tau\leq t\}\) is the external
ellipsoidal tube by the instant of time.
+where \({\mathcal E}^{U}_{+}[t,l]=\underset{\tau}{\bigcup}\{{\mathcal
E}(\overline{x}(\tau),X_{+}(\tau,l))|t_0\leq\tau\leq t\}\) is the external
ellipsoidal tube by instant of time.

-Similar approxiamtion can be calculated with the internal ellipsoidal tube
by the instant of time \({\mathcal
E}^{U}_{-}[t,l]=\underset{\tau}{\bigcup}\{{\mathcal
E}(\overline{x}(\tau),X_{-}(\tau,l))|t_0\leq\tau\leq t\}\):
+Similar approxiamtion can be calculated with the internal ellipsoidal tube
by instant of time \({\mathcal
E}^{U}_{-}[t,l]=\underset{\tau}{\bigcup}\{{\mathcal
E}(\overline{x}(\tau),X_{-}(\tau,l))|t_0\leq\tau\leq t\}\):
\begin{gather}
\begin{split}{\mathcal X}^{U}[t]\supseteq\underset{l}{\bigcup}\{{\mathcal
E}^{U}_{-}[t,l]) |l\in {\mathcal S}_1(0)\}.\end{split}\notag
\end{gather}
-Note that in general case ellipsoidal tube \({\mathcal E}^{U}_{+}[t,l]\)
is not tight approximation.
+Note that in general case ellipsoidal tube \({\mathcal E}^{U}_{+}[t,l]\)
is not tight approximation. For more information see
{\hyperref[chap_ellTube:gag2012]{{[}GAG2012{]}}}.

So, all in all, there are two types of ellipsoid tube objects that we can
work with using \emph{Ellipsoidal Toolbox}:
\begin{itemize}
@@ -5363,12 +5365,12 @@
ellipsoidal tubes that are described in
\emph{gras.ellapx.smartdb.rels.EllTube} class;

\item {}
-tubes by the instant of time described in
\emph{gras.ellapx.smartdb.rels.EllUnionTube} class
+tubes by instant of time described in
\emph{gras.ellapx.smartdb.rels.EllUnionTube} class
(see {\hyperref[chap_implement:union-label]{\emph{formula}}}).

\end{itemize}

-These two type of objects can be projected on specified subspaces. The
projections of ellipsoid tubes can be either static or dynamic. The are
described in gras.ellapx.smartdb.rels.EllTubeProj class. As for the tubes
by the instant of time, they can only be projected on static subspaces.
These projections are described in
gras.ellapx.smartdb.rels.EllUnionTubeStaticProj class. For more information
about these types of projections and their differences and for examples see
this {\hyperref[chap_implement:section-label]{\emph{link}}}.
+These two type of objects can be projected on specified subspaces. The
projections of ellipsoid tubes can be either static or dynamic. The are
described in gras.ellapx.smartdb.rels.EllTubeProj class. As for the tubes
by instant of time, they can only be projected on static subspaces. These
projections are described in
gras.ellapx.smartdb.rels.EllUnionTubeStaticProj class. For more information
about these types of projections and their differences and for examples see
this {\hyperref[chap_implement:section-label]{\emph{link}}}.

\section{Ellipsoid tubes}
@@ -6098,9 +6100,9 @@
\end{Verbatim}

-\section{Tubes by the instant of time}
-\label{chap_ellTube:tubes-by-the-instant-of-time}
-As with ellipsoid tube objects there are several methods that we can use
while working with tubes by the instant of time. First of all we can create
tubes by the instant of time using \emph{fromEllTubes} method:
+\section{Tubes by instant of time}
+\label{chap_ellTube:tubes-by-instant-of-time}
+As with ellipsoid tube objects, there are several methods that we can use
while working with tubes by instant of time. First of all we can create
tubes by instant of time using \emph{fromEllTubes} method:

\begin{Verbatim}[commandchars=\\\{\},numbers=left,firstnumber=1,stepnumber=1]
\PYG{c}{\PYGZpc{} An example of creating EllUnionTube object using
FROMELLTUBES method.}
@@ -6119,7 +6121,7 @@

\PYG{n}{gras}\PYG{p}{.}\PYG{n}{ellapx}\PYG{p}{.}\PYG{n}{smartdb}\PYG{p}{.}\PYG{n}{rels}\PYG{p}{.}\PYG{n}{EllUnionTube}\PYG{p}{.}\PYG{n}{fromEllTubes}\PYG{p}{(}\PYG{n}{ellTubeObj}\PYG{p}{)}\PYG{p}{;}
\end{Verbatim}

-From here on we will use the \emph{getUnion} function so we can get a tube
by the instant of time and work with it further on. As we have created a
tubes by the instant of time object, we can get all the types of differet
data about it. There is a set of methods that can give information about
the data stored in the object and give access to it.
+From here on we will use the \emph{getUnion} function so we can get a tube
by instant of time and work with it further on. As we have created a tubes
by instant of time object, we can get all the types of differet data about
it. There is a set of methods that can give information about the data
stored in the object and give access to it.

\begin{Verbatim}[commandchars=\\\{\},numbers=left,firstnumber=1,stepnumber=1]
\PYG{c}{\PYGZpc{} An example of GETDATA method\PYGZsq{}s usage.}
@@ -6269,7 +6271,7 @@

\PYG{n}{ellUnionObj}\PYG{p}{.}\PYG{n}{display}\PYG{p}{(}\PYG{p}{)}\PYG{p}{;}
\end{Verbatim}

-Also we can compare tubes by the instant of time using \emph{isEqual}
method.
+Also we can compare tubes by instant of time using \emph{isEqual} method.

\begin{Verbatim}[commandchars=\\\{\},numbers=left,firstnumber=1,stepnumber=1]
\PYG{c}{\PYGZpc{} An example of ISEQUAL method usage. The compared
ellTubeUnions are not}
@@ -6280,7 +6282,7 @@
\PYG{n}{res} \PYG{p}{=}
\PYG{n}{firstUnionObj}\PYG{p}{.}\PYG{n}{isEqual}\PYG{p}{(}\PYG{n}{secondUnionObj}\PYG{p}{)}\PYG{p}{;}
\end{Verbatim}

-At last, as it has already been said tubes by the instant of time can be
projected only on static subspaces. It can be done in two ways.
+At last, as it has already been said tubes by instant of time can be
projected only on static subspaces. It can be done in two ways.

\begin{Verbatim}[commandchars=\\\{\},numbers=left,firstnumber=1,stepnumber=1]
\PYG{c}{\PYGZpc{} An example of calculating EllTubeUnion object\PYGZsq{}s
projection using PROJECT}
@@ -6303,7 +6305,7 @@
\PYG{n}{statEllTubeProjObj} \PYG{p}{=}
\PYG{n}{unionEllTubeObj}\PYG{p}{.}\PYG{n}{projectStatic}\PYG{p}{(}\PYG{n}{projMatList}\PYG{p}{)}\PYG{p}{;}
\end{Verbatim}

-As for ellipsoid tube projections, a special function is used to create
the projection matrix for tubes by the instant of time:
+As for ellipsoid tube projections, a special function is used to create
the projection matrix for tubes by instant of time:

\begin{Verbatim}[commandchars=\\\{\},numbers=left,firstnumber=1,stepnumber=1]
function [projOrthMatArray,projOrthMatTransArray]=...
@@ -6315,21 +6317,21 @@
\end{Verbatim}

-\section{Projections of ellipsoid tubes and unions}
-\label{chap_ellTube:projections-of-ellipsoid-tubes-and-unions}
-As it has already been said we can create either static or dynamic
projections for ellipsoid tubes and only static projections for tubes by
the instant of time. There are several methods in \emph{Ellipsoidal
Toolbox} for that. Most of them has already been described:
+\section{Projections of ellipsoid tubes and tubes by instant of time}
+\label{chap_ellTube:projections-of-ellipsoid-tubes-and-tubes-by-instant-of-time}
+As it has already been said we can create either static or dynamic
projections for ellipsoid tubes and only static projections for tubes by
instant of time. There are several methods in \emph{Ellipsoidal Toolbox}
for that. Most of them has already been described:
\begin{itemize}
\item {}
\emph{project}, \emph{projectStatic} and \emph{projectToOrths} for
ellipsoid tubes;

\item {}
-\emph{project} and \emph{projectStatic} for tubes by the instant of time.
+\emph{project} and \emph{projectStatic} for tubes by instant of time.

\end{itemize}

-It should be mentioned that from here on all the examples are written for
ellipsoid tube projections, but their usage is the same for the projections
of tubes by the instant of time. We wiil use \emph{getProj} function to
create ellipsoid tube projection that we will work with.
+It should be mentioned that from here on all the examples are written for
ellipsoid tube projections, but their usage is the same for the projections
of tubes by instant of time. We wiil use \emph{getProj} function to create
ellipsoid tube projection that we will work with.

-As with ellipsoid tubes and tubes by the instant of time we can get all
the types of differet data about projections. There is a set of methods
that can give information about the data stored in the object and give
access to it.
+As with ellipsoid tubes and tubes by instant of time we can get all the
types of differet data about projections. There is a set of methods that
can give information about the data stored in the object and give access to
it.

\begin{Verbatim}[commandchars=\\\{\},numbers=left,firstnumber=1,stepnumber=1]
\PYG{c}{\PYGZpc{} An example of GETDATA method\PYGZsq{}s usage.}
@@ -18085,6 +18087,9 @@
Estimation with the Cell Transmission Model. In \emph{Proceedings of the
American Control Conference}, 3750–3755. Denver, Colorado, USA.
}
+\bibitem[GAG2012]{GAG2012}{\phantomsection\label{chap_ellTube:gag2012}
+Gagarinov P.V. 2012. Computation of Alternated Reachability Tubes for
Linear Control Systems under Uncertainty. \emph{Moscow University
Computational Mathematics and Cybernetics}, 2012, Vol. 36, No. 4, pp. 169–
177.
+}
\bibitem[VER2001]{VER2001}{\phantomsection\label{chap_summary:ver2001}
Veres, S. M., A. V. Kuntsevich, I. V. Vályi, S. Hermsmeyer, and D. S.
Wall. 2001. Geometric Bounding Toolbox for MATLAB. \emph{MATLAB/Simulink
=======================================
--- /branches/issue_133_aatanesyan/doc/chap_ellTube.rst Wed Mar 5 22:30:52
2014 UTC
+++ /branches/issue_133_aatanesyan/doc/chap_ellTube.rst Thu Mar 6 15:09:17
2014 UTC
@@ -1,35 +1,37 @@
-Ellipsoid tubes, tubes by the instant of time and their projections
+Ellipsoid tubes, tubes by instant of time and their projections
===================================================================

-**Definition.** For any matrix asimptotic monotone
function :math:`M(\cdot)`, which defines the configuration of the
environment of any point of time, the quadratically regularized alternated
reach set of system with disturbance is
+**Definition.** For any regularization matrix :math:`M(\cdot)` the
quadratically regularized alternated reach set of system with uncertainty is

.. math::
{\mathcal X}^{q}_{U}(t,t_0,{\mathcal X}^{0},M(\cdot)) =
\underset{t_0\leq\tau\leq t}{\bigcup}{\mathcal
X}^{q}_{U}(\tau,t_0,{\mathcal X}^{0},M (\cdot)).

-Identify as :math:`{\mathcal E}(\overline{x}(t),X_{+}(t,l))`
и :math:`{\mathcal E}(\overline{x}(t),X_{-}(t,l))` the tight external and
internal approximations along :math:`l(\cdot)` good direction such
as :math:`l(t_0)=l`. Then the reach set by instant of time can be described
as
+Note that if system doesn't have uncertainty then :math:`M(\cdot)=0`.
+
+By :math:`{\mathcal E}(\overline{x}(t),X_{+}(t,l))` and :math:`{\mathcal
E}(\overline{x}(t),X_{-}(t,l))` denote tight external and internal
approximations along :math:`l(\cdot)` good direction such
as :math:`l(t_0)=l`. Then the reach set by instant of time can be described
as

.. math::
{\mathcal X}^{q}_{U}[t]=\underset{\tau}{\bigcup}\underset{l}{\bigcap}\{
{\mathcal E}(\overline{x}(\tau),X_{+}(\tau,l)) |l\in {\mathcal
S}_1(0), \tau\leq t\}\subseteq\underset{l} {\bigcap}\{{\mathcal
E}^{U}_{+}[t,l]) |l\in {\mathcal S}_1(0)\},

-where :math:`{\mathcal E}^{U}_{+}[t,l]=\underset{\tau}{\bigcup}\{{\mathcal
E}(\overline{x}(\tau),X_{+}(\tau,l))|t_0\leq\tau\leq t\}` is the external
ellipsoidal tube by the instant of time.
+where :math:`{\mathcal E}^{U}_{+}[t,l]=\underset{\tau}{\bigcup}\{{\mathcal
E}(\overline{x}(\tau),X_{+}(\tau,l))|t_0\leq\tau\leq t\}` is the external
ellipsoidal tube by instant of time.

-Similar approxiamtion can be calculated with the internal ellipsoidal tube
by the instant of time :math:`{\mathcal
E}^{U}_{-}[t,l]=\underset{\tau}{\bigcup}\{{\mathcal
E}(\overline{x}(\tau),X_{-}(\tau,l))|t_0\leq\tau\leq t\}`:
+Similar approxiamtion can be calculated with the internal ellipsoidal tube
by instant of time :math:`{\mathcal
E}^{U}_{-}[t,l]=\underset{\tau}{\bigcup}\{{\mathcal
E}(\overline{x}(\tau),X_{-}(\tau,l))|t_0\leq\tau\leq t\}`:

.. math::
{\mathcal X}^{U}[t]\supseteq\underset{l}{\bigcup}\{{\mathcal
E}^{U}_{-}[t,l]) |l\in {\mathcal S}_1(0)\}.

-Note that in general case ellipsoidal tube :math:`{\mathcal
E}^{U}_{+}[t,l]` is not tight approximation.
+Note that in general case ellipsoidal tube :math:`{\mathcal
E}^{U}_{+}[t,l]` is not tight approximation. For more information see
[GAG2012]_.

So, all in all, there are two types of ellipsoid tube objects that we can
work with using *Ellipsoidal Toolbox*:

- ellipsoidal tubes that are described in
*gras.ellapx.smartdb.rels.EllTube* class;

-- tubes by the instant of time described in
*gras.ellapx.smartdb.rels.EllUnionTube* class
+- tubes by instant of time described in
*gras.ellapx.smartdb.rels.EllUnionTube* class
(see :ref:`formula <union-label>`).

-These two type of objects can be projected on specified subspaces. The
projections of ellipsoid tubes can be either static or dynamic. The are
described in gras.ellapx.smartdb.rels.EllTubeProj class. As for the tubes
by the instant of time, they can only be projected on static subspaces.
These projections are described in
gras.ellapx.smartdb.rels.EllUnionTubeStaticProj class. For more information
about these types of projections and their differences and for examples see
this :ref:`link <section-label>`.
+These two type of objects can be projected on specified subspaces. The
projections of ellipsoid tubes can be either static or dynamic. The are
described in gras.ellapx.smartdb.rels.EllTubeProj class. As for the tubes
by instant of time, they can only be projected on static subspaces. These
projections are described in
gras.ellapx.smartdb.rels.EllUnionTubeStaticProj class. For more information
about these types of projections and their differences and for examples see
this :ref:`link <section-label>`.

Ellipsoid tubes
---------------
@@ -195,16 +197,16 @@
:language: matlab
:linenos:

-Tubes by the instant of time
-----------------------------
+Tubes by instant of time
+------------------------

-As with ellipsoid tube objects there are several methods that we can use
while working with tubes by the instant of time. First of all we can create
tubes by the instant of time using *fromEllTubes* method:
+As with ellipsoid tube objects, there are several methods that we can use
while working with tubes by instant of time. First of all we can create
tubes by instant of time using *fromEllTubes* method:

..
literalinclude:: ../products/+gras/+ellapx/+smartdb/+test/+examples//example_fromEllTubes.m
:language: matlab
:linenos:

-From here on we will use the *getUnion* function so we can get a tube by
the instant of time and work with it further on. As we have created a tubes
by the instant of time object, we can get all the types of differet data
about it. There is a set of methods that can give information about the
data stored in the object and give access to it.
+From here on we will use the *getUnion* function so we can get a tube by
instant of time and work with it further on. As we have created a tubes by
instant of time object, we can get all the types of differet data about it.
There is a set of methods that can give information about the data stored
in the object and give access to it.

..
literalinclude:: ../products/+gras/+ellapx/+smartdb/+test/+examples//example_getDataUnion.m
:language: matlab
@@ -225,13 +227,13 @@
:language: matlab
:linenos:

-Also we can compare tubes by the instant of time using *isEqual* method.
+Also we can compare tubes by instant of time using *isEqual* method.

..
literalinclude:: ../products/+gras/+ellapx/+smartdb/+test/+examples//example_isEqualUnion.m
:language: matlab
:linenos:

-At last, as it has already been said tubes by the instant of time can be
projected only on static subspaces. It can be done in two ways.
+At last, as it has already been said tubes by instant of time can be
projected only on static subspaces. It can be done in two ways.

..
literalinclude:: ../products/+gras/+ellapx/+smartdb/+test/+examples//example_project.m
:language: matlab
@@ -241,28 +243,28 @@
:language: matlab
:linenos:

-As for ellipsoid tube projections, a special function is used to create
the projection matrix for tubes by the instant of time:
+As for ellipsoid tube projections, a special function is used to create
the projection matrix for tubes by instant of time:

..
literalinclude:: ../products/+gras/+ellapx/+smartdb/+test/+examples//fGetProjMat.m
:language: matlab
:linenos:

-Projections of ellipsoid tubes and unions
------------------------------------------
+Projections of ellipsoid tubes and tubes by instant of time
+-----------------------------------------------------------

-As it has already been said we can create either static or dynamic
projections for ellipsoid tubes and only static projections for tubes by
the instant of time. There are several methods in *Ellipsoidal Toolbox* for
that. Most of them has already been described:
+As it has already been said we can create either static or dynamic
projections for ellipsoid tubes and only static projections for tubes by
instant of time. There are several methods in *Ellipsoidal Toolbox* for
that. Most of them has already been described:

- *project*, *projectStatic* and *projectToOrths* for ellipsoid tubes;

-- *project* and *projectStatic* for tubes by the instant of time.
+- *project* and *projectStatic* for tubes by instant of time.

-It should be mentioned that from here on all the examples are written for
ellipsoid tube projections, but their usage is the same for the projections
of tubes by the instant of time. We wiil use *getProj* function to create
ellipsoid tube projection that we will work with.
+It should be mentioned that from here on all the examples are written for
ellipsoid tube projections, but their usage is the same for the projections
of tubes by instant of time. We wiil use *getProj* function to create
ellipsoid tube projection that we will work with.

..
literalinclude:: ../products/+gras/+ellapx/+smartdb/+test/+examples//example_getProj.m
:language: matlab
:linenos:

-As with ellipsoid tubes and tubes by the instant of time we can get all
the types of differet data about projections. There is a set of methods
that can give information about the data stored in the object and give
access to it.
+As with ellipsoid tubes and tubes by instant of time we can get all the
types of differet data about projections. There is a set of methods that
can give information about the data stored in the object and give access to
it.

..
literalinclude:: ../products/+gras/+ellapx/+smartdb/+test/+examples//example_getDataProj.m
@@ -307,3 +309,4 @@

To read more about the differences of these projections :ref:`goto
<goto-label>`. Also there you can find some illustrations for both ellTube
projections and ellUnionTube projections.

+.. [GAG2012] Gagarinov P.V. 2012. Computation of Alternated Reachability
Tubes for Linear Control Systems under Uncertainty. *Moscow University
Computational Mathematics and Cybernetics*, 2012, Vol. 36, No. 4, pp. 169–
177.
=======================================
--- /branches/issue_133_aatanesyan/doc/chap_implement.rst Wed Mar 5
02:41:50 2014 UTC
+++ /branches/issue_133_aatanesyan/doc/chap_implement.rst Thu Mar 6
15:09:17 2014 UTC
@@ -405,6 +405,7 @@
.. _union-label:

.. math::
+ :label: ellUnion

{\mathcal X}_{U}[t]=\bigcup \limits_{\tau\leqslant t}{\mathcal X}[\tau],

=======================================
--- /branches/issue_133_aatanesyan/doc/conf.py Tue Dec 10 16:34:21 2013 UTC
+++ /branches/issue_133_aatanesyan/doc/conf.py Thu Mar 6 15:09:17 2014 UTC
@@ -58,7 +58,7 @@
# The short X.Y version.
version = '2.0'
# The full version, including alpha/beta/rc tags.
-release = '2.0 beta 1'
+release = '2.0.1'

# The language for content autogenerated by Sphinx. Refer to documentation
# for a list of supported languages.
=======================================
--- /branches/issue_133_aatanesyan/doc/main_elltoolmanual.tex Tue Dec 10
16:24:10 2013 UTC
+++ /branches/issue_133_aatanesyan/doc/main_elltoolmanual.tex Thu Mar 6
15:09:17 2014 UTC
@@ -24,7 +24,7 @@
ELLIPSOIDAL TOOLBOX%\thanks{Research supported by NSF Grant CCR-00225610.}
\\
%Manual DRAFT\\
-\normalsize{ver. 2.0 beta 1}
+\normalsize{ver. 2.0.1}
\author{Peter Gagarinov, Alex A. Kurzhanskiy,}
\date{2013}
}

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