diff --git a/doc/manual/manual/ellipsoids/Ellipsoid_class.rst b/doc/manual/manual/ellipsoids/Ellipsoid_class.rst
index 5b9eb2f7e..cf149374b 100644
--- a/doc/manual/manual/ellipsoids/Ellipsoid_class.rst
+++ b/doc/manual/manual/ellipsoids/Ellipsoid_class.rst
@@ -10,43 +10,63 @@ This page describes the Ellipsoid classes used in Codac 2 as well as some functi
Class Ellipsoid
---------------
-As presented in :ref:what_is_ellipsoids the n-dimensional ellipsoids is defined by a center point and a shape matrix
+The Ellipsoid class can be used to declare a n-dimensional ellipsoid.
+As presented in :ref:what_is_ellipsoids the ellipsoid is defined by a center point and a shape matrix.
+Ellipsoids can be drawn in VIBES via the .draw_ellipsoid function, as illustrated by :ref:Fig.
+
.. tabs::
.. code-tab:: py
- # Init drawing figure
- fig1 = Figure2D('Linear and nonlinear mappings', GraphicOutput.VIBES)
- fig1.set_axes(axis(0, [0, 1.5]), axis(1, [-1., 0.5]))
- fig1.set_window_properties([0, 100], [500, 500])
+ # Init drawing figure
+ fig1 = Figure2D('Linear and nonlinear mappings', GraphicOutput.VIBES)
+ fig1.set_axes(axis(0, [-0.1, 1.3]), axis(1, [-1.2, 0.2]))
+ fig1.set_window_properties([0, 100], [500, 500])
- mu = Vector([1., 0.])
- G = Matrix([[0.05, 0.0],
- [0., 0.05]])
- e1 = Ellipsoid(mu, G)
+ mu = Vector([1., 0.])
+ G = Matrix([[0.05, 0.0],
+ [0., 0.05]])
+ e1 = Ellipsoid(mu, G)
- fig1.draw_ellipsoid(e1, [Color.red(), Color.red(0.3)]) # drawing
+ fig1.draw_ellipsoid(e1, [Color.red(), Color.red(0.3)]) # drawing
.. code-tab:: c++
- // Init drawing figure
- Figure2D fig1("Linear and nonlinear mappings", GraphicOutput::VIBES);
- fig1.set_axes(axis(0, {0, 1.5}), axis(1, {-1., 0.5}));
- fig1.set_window_properties({0, 100}, {500, 500});
-
- Vector mu({1., 0.}); // center point
- Matrix G({{0.05, 0.0},{0., 0.05}}); // shape matrix
- Ellipsoid e1(mu,G); // ellipsoid
-
- fig1.draw_ellipsoid(e1, {Color::red(), Color::red(0.3)}); // draw
+ // Init drawing figure
+ Figure2D fig1("Linear and nonlinear mappings", GraphicOutput::VIBES);
+ fig1.set_axes(axis(0, {-0.1, 1.3}), axis(1, {-1.2, 0.2}));
+ fig1.set_window_properties({0, 100}, {500, 500});
+
+ Vector mu({1., 0.}); // center point
+ Matrix G({{0.05, 0.0},{0., 0.05}}); // shape matrix
+ Ellipsoid e1(mu,G); // ellipsoid
+
+ fig1.draw_ellipsoid(e1, {Color::red(), Color::red(0.3)}); // draw
+ cout << e1 << endl;
+ /*
+ Ellipsoid 2d:
+ mu=[ 1 ; 0 ]
+ G=
+ [[ 0.05 , 0 ]
+ [ 0 , 0.05 ]]
+ */
Linear and nonlinear mappings
-----------------------------
-.. figure:: codaca.png
+Linear and nonlinear mappings can be applied on ellipsoids.
+
+For every ellipsoid $ex$, square matrix $A$ and vector $b$,
+the function *unreliable_linear_mapping* compute the ellipsoid $ey=A\cdot ex + b$.
+
+Nonlinear mappings can be declared with the AnalyticFunction class. For every ellipsoid $ex$ and nonlinear mapping $h$,
+the function nonlinear_mapping compute an ellipsoid $ey$ that enclose the image of $ex$ by $h$ such that
+$h(ex)\in ey$
+
+.. figure:: linear_and_nonlinear_mappings.png
:width: 400
- Figure 1 - caption
+ Figure 1 - Illustration of successive linear and nonlinear mappings on ellipsoids
.. tabs::
@@ -98,6 +118,16 @@ Linear and nonlinear mappings
Projection of the ellipsoids
----------------------------
+High dimensional ellipsoids can illustrated with 2D projection.
+In Codac, the projection is made by the Figure2D object via the .draw_ellipsoid function:
+the ellipsoid is projected on the plane (0,i,j),
+where the axis i and j are specified via the .set_axes function of the figure.
+
+.. figure:: projections.png
+ :width: 800
+
+ Figure 2 - Projections of the ellipsoids $e4$, $e5$ and $e6$ in the 3D XYZ space
+
.. tabs::
.. code-tab:: py
@@ -185,31 +215,22 @@ Projection of the ellipsoids
fig3.draw_ellipsoid(e6, {Color::green(), Color::green(0.3)});
fig4.draw_ellipsoid(e6, {Color::green(), Color::green(0.3)});
-.. figure:: codacb.png
- :width: 400
-
- Figure 2 - caption
-
-.. figure:: codace.png
- :width: 400
-
- Figure 3 - caption
-
-.. figure:: codacf.png
- :width: 400
-
- Figure 4 - caption
Inclusion tests
---------------
+
+The function .is_concentric_subset can test if two concentric ellipsoids are strictly included in each other.
+The function return a BoolInterval that can be: [ true ] if the inclusion is verified /
+[ true, false ] if the method is not able to conclude
+
.. tabs::
.. code-tab:: py
print('\nInclusion test e5 in e4: ', e5.is_concentric_subset(e4))
- print('\nclusion test e4 in e5: ', e4.is_concentric_subset(e5))
- print('\nclusion test e4 in e6: ', e6.is_concentric_subset(e4))
- print('\nclusion test e5 in e6: ', e5.is_concentric_subset(e6))
+ print('\nInclusion test e4 in e5: ', e4.is_concentric_subset(e5))
+ print('\nInclusion test e4 in e6: ', e6.is_concentric_subset(e4))
+ print('\nInclusion test e5 in e6: ', e5.is_concentric_subset(e6))
.. code-tab:: c++
@@ -218,13 +239,25 @@ Inclusion tests
cout << "Inclusion test e4 in e6: " << e6.is_concentric_subset(e4) << endl;
cout << "Inclusion test e5 in e6: " << e5.is_concentric_subset(e6) << endl;
+ /*
+ Inclusion test e5 in e4: [ true ] -> e5 is included in e4
+ Inclusion test e4 in e5: [ true, false ] -> not able to conclude
+ Inclusion test e4 in e6: [ true, false ]
+ Inclusion test e5 in e6: [ true, false ]
+ */
+
+
Degenerated Ellipsoids & singular mappings
------------------------------------------
-.. figure:: codace.png
- :width: 400
+It is also possible to have Degenerated Ellipsoids and to apply singular mappings on ellipsoids.
+There functionalities are handled by the Ellipsoid class and the nonlinear_mapping function
+
+.. figure:: singular_case.png
+ :width: 600
- Figure 5 - caption
+ Figure 3 - Singular cases. $e11$ is the image of $e9$ by the nonlinear mapping $h2$.
+ $e12$ is the image of $e10$ by the nonlinear mapping $h3$
.. tabs::
@@ -293,14 +326,46 @@ Degenerated Ellipsoids & singular mappings
cout << "\nNon-degenerate ellipsoid e10 (red):\n" << e10 << endl;
cout << "\nImage of singular mapping e12 (green):\n" << e12 << endl;
+ /*
+ Degenerate ellipsoid e9 (blue):
+ Ellipsoid 2d:
+ mu=[ 0 ; 0.5 ]
+ G=
+ [[ 0.25 , 0 ]
+ [ 0 , 0 ]]
+
+ Image of degenerated ellipsoid e11 (green):
+ Ellipsoid 2d:
+ mu=[ 1 ; 0.5 ]
+ G=
+ [[ 0.500005 , 0.00310879 ]
+ [ -0.250002 , 0.00621758 ]]
+
+ Non-degenerate ellipsoid e10 (red):
+ Ellipsoid 2d:
+ mu=[ 0 ; -0.5 ]
+ G=
+ [[ 0.25 , 0 ]
+ [ 0 , 0.25 ]]
+
+ Image of singular mapping e12 (green):
+ Ellipsoid 2d:
+ mu=[ 1 ; -0.5 ]
+ G=
+ [[ 0.698771 , 2.20794e-16 ]
+ [ 0.698771 , -2.20794e-16 ]]
+ */
+
Stability analysis
------------------
-.. figure:: codacc.png
- :width: 400
+The function stability_analysis can compute a bassin of attraction for discrete time systems.
+
+.. figure:: stability.png
+ :width: 600
- Figure 6 - caption
+ Figure 6 - Stability analysis for the discrete system $\boldsymbol{x}_{k+1} = \boldsymbol{h4}\left(\boldsymbol{x}_k\right)$
.. tabs::
@@ -342,6 +407,22 @@ Stability analysis
fig6.draw_ellipsoid(e13, {Color::red(), Color::red(0.3)});
fig6.draw_ellipsoid(e13_out, {Color::green(), Color::green(0.3)});
+ /*
+ Stability analysis: the system is stable
+ Ellipsoidal domain of attraction e13 (red):
+ Ellipsoid 2d:
+ mu=[ 0 ; 0 ]
+ G=
+ [[ 0.0530036 , -0.0162023 ]
+ [ -0.0162023 , 0.0599474 ]]
+ Outter enclosure e13_out of the Image of e13 by h4 (green):
+ Ellipsoid 2d:
+ mu=[ 0 ; 2.4895e-17 ]
+ G=
+ [[ 0.0449537 , 0.0137871 ]
+ [ -0.0346424 , 0.0381183 ]]
+ */
+
diff --git a/doc/manual/manual/ellipsoids/codaca.png b/doc/manual/manual/ellipsoids/codaca.png
deleted file mode 100644
index 829096f27..000000000
Binary files a/doc/manual/manual/ellipsoids/codaca.png and /dev/null differ
diff --git a/doc/manual/manual/ellipsoids/codacb.png b/doc/manual/manual/ellipsoids/codacb.png
deleted file mode 100644
index 0b59a063a..000000000
Binary files a/doc/manual/manual/ellipsoids/codacb.png and /dev/null differ
diff --git a/doc/manual/manual/ellipsoids/codacc.png b/doc/manual/manual/ellipsoids/codacc.png
deleted file mode 100644
index 903dada76..000000000
Binary files a/doc/manual/manual/ellipsoids/codacc.png and /dev/null differ
diff --git a/doc/manual/manual/ellipsoids/codacd.png b/doc/manual/manual/ellipsoids/codacd.png
deleted file mode 100644
index dad978234..000000000
Binary files a/doc/manual/manual/ellipsoids/codacd.png and /dev/null differ
diff --git a/doc/manual/manual/ellipsoids/codace.png b/doc/manual/manual/ellipsoids/codace.png
deleted file mode 100644
index e914eedbd..000000000
Binary files a/doc/manual/manual/ellipsoids/codace.png and /dev/null differ
diff --git a/doc/manual/manual/ellipsoids/codacf.png b/doc/manual/manual/ellipsoids/codacf.png
deleted file mode 100644
index fcabf76ee..000000000
Binary files a/doc/manual/manual/ellipsoids/codacf.png and /dev/null differ
diff --git a/doc/manual/manual/ellipsoids/linear_and_nonlinear_mappings.png b/doc/manual/manual/ellipsoids/linear_and_nonlinear_mappings.png
new file mode 100644
index 000000000..c96c65734
Binary files /dev/null and b/doc/manual/manual/ellipsoids/linear_and_nonlinear_mappings.png differ
diff --git a/doc/manual/manual/ellipsoids/projections.png b/doc/manual/manual/ellipsoids/projections.png
new file mode 100644
index 000000000..ba12683f5
Binary files /dev/null and b/doc/manual/manual/ellipsoids/projections.png differ
diff --git a/doc/manual/manual/ellipsoids/singular_case.png b/doc/manual/manual/ellipsoids/singular_case.png
new file mode 100644
index 000000000..41fd8a928
Binary files /dev/null and b/doc/manual/manual/ellipsoids/singular_case.png differ
diff --git a/doc/manual/manual/ellipsoids/stability.png b/doc/manual/manual/ellipsoids/stability.png
new file mode 100644
index 000000000..ede4277c8
Binary files /dev/null and b/doc/manual/manual/ellipsoids/stability.png differ
diff --git a/examples/ellipsoid_example/main.cpp b/examples/ellipsoid_example/main.cpp
index 46683469c..34c1fb3fc 100644
--- a/examples/ellipsoid_example/main.cpp
+++ b/examples/ellipsoid_example/main.cpp
@@ -10,7 +10,7 @@ int main() {
cout << "Linear and nonlinear mappings:" << endl;
Figure2D fig1("Linear and nonlinear mappings", GraphicOutput::VIBES);
- fig1.set_axes(axis(0, {0, 1.5}), axis(1, {-1., 0.5}));
+ fig1.set_axes(axis(0, {-0.1, 1.3}), axis(1, {-1.2, 0.2}));
fig1.set_window_properties({0, 100}, {500, 500});
// initial ellipsoid
@@ -89,9 +89,10 @@ int main() {
fig3.set_window_properties({1200, 100}, {500, 500});
fig4.set_window_properties({0, 600}, {500, 500});
- fig2.set_axes(axis(0, {-3, 3}), axis(1, {-3, 3}));
- fig3.set_axes(axis(1, {-3, 3}), axis(2, {-3, 3}));
- fig4.set_axes(axis(0, {-3, 3}), axis(2, {-3, 3}));
+ fig2.set_axes(axis(0, {-3.1, 3.1}), axis(1, {-3.1, 3.1}));
+ fig2.set_axes(axis(0, {-3.1, 3.1}), axis(1, {-3.1, 3.1}));
+ fig3.set_axes(axis(1, {-3.1, 3.1}), axis(2, {-3.1, 3.1}));
+ fig4.set_axes(axis(0, {-3.1, 3.1}), axis(2, {-3.1, 3.1}));
fig2.draw_ellipsoid(e4, {Color::blue(), Color::blue(0.3)});
fig3.draw_ellipsoid(e4, {Color::blue(), Color::blue(0.3)});
diff --git a/examples/ellipsoid_example/main.py b/examples/ellipsoid_example/main.py
index c349dd933..69212b92c 100644
--- a/examples/ellipsoid_example/main.py
+++ b/examples/ellipsoid_example/main.py
@@ -5,7 +5,7 @@
# ----------------------------------------------------------
fig1 = Figure2D('Linear and nonlinear mappings', GraphicOutput.VIBES)
-fig1.set_axes(axis(0, [0, 1.5]), axis(1, [-1., 0.5]))
+fig1.set_axes(axis(0, [-0.1, 1.3]), axis(1, [-1.2, 0.2]))
fig1.set_window_properties([0, 100], [500, 500])
# initial ellipsoid