issue #54

improved indices

doc/chap_examples.rst

@@ -520,14 +520,23 @@ are shown in green. Observe that in the congested mode, the density
 the guard is not actually reached, because the state evolves according
 to the green region.
 
+
+.. raw:: html
+
+	<h2>References</h2>
+
+.. [SUN2003] L.Muñoz, X.Sun, R.Horowitz, and L.Alvarez. 2003. Traffic Density
+   Estimation with the Cell Transmission Model. In *Proceedings of the
+   American Control Conference*, 3750–3755. Denver, Colorado, USA.
+      
 Inverted Pendulum On a Cart
 ---------------------------
 
 .. _invpendfig:
 
 .. figure:: /pic/chapter06_section05_inpendulum.png
    :alt: invpend
-   :width: 50 %
+   :width: 30 %
 
    Inverted pendulum
 
@@ -541,7 +550,7 @@ Therefore we define
 
 -  :math:`L` - length of the rod;
 
--  :math:`k_1, k_2` - the coefficient of dynamic viscosity for the bodies 1 and 2 respectively;
+-  :math:`k_1, k_2` - the coefficient of viscous friction for the bodies 1 and 2 respectively;
 
 -  :math:`J_c` - inertia moment of the second body relative to its center of mass;
 
@@ -553,12 +562,10 @@ Firstly we consider the system without any control. Writing down expressions for
 .. math::
    :label: invpend_energy
 
-   \begin{aligned}
-   T_1 &= \frac{m_1v^2}{2}, \\
-   T_2 &= \frac{J_c\omega^2}{2}, \\
-   \Pi_1 &= 0, \\
-   \Pi_2 &= m_2g\frac{L}{2}\sin(\theta). 
-   \end{aligned}
+   T_1 = \frac{m_1v^2}{2},
+   T_2 = \frac{J_c\omega^2}{2},
+   \Pi_1 = 0,
+   \Pi_2 = m_2g\frac{L}{2}\sin(\theta). 
 
 Here :math:`T_1` and :math:`T_2` stand for kinetic energy of cart and rod respectively, :math:`\Pi_1` and :math:`\Pi_2` stand for potential energy, :math:`v` is a speed of a cart and :math:`\omega` is an angular velocity of the rod. Further we will replace :math:`v` with :math:`\dot x`, :math:`\omega` with :math:`\dot \theta`. 
 
@@ -567,10 +574,8 @@ As there is viscosity, we write down generalized forces:
 .. math::
    :label: invpend_forces
 
-   \begin{aligned}
-   Q_1 &= k_1\dot x, \\
-   Q_2 &= k_2L\dot \theta.
-   \end{aligned}
+   Q_1 = k_1\dot x,
+   Q_2 = k_2L\dot \theta.
 
 Writing down the Langrange equations in case of potential and nonpotential forces we obtain:
 
@@ -579,7 +584,7 @@ Writing down the Langrange equations in case of potential and nonpotential force
 
    \frac{d}{dt}(\frac{\partial T}{\partial \dot q_i}) - \frac{\partial T}{\partial q_i} + \frac{\partial \Pi}{\partial q_i} = Q_i.
 
-Here the Lagrange coordinates :math:`q_1 = x, q_2 = \theta`, :math:`T = T_1 + T_2, \Pi = \Pi_1 + \Pi_2`.
+Here the Lagrange coordinates :math:`q_1 = x, q_2 = \theta`.
 
 Applying previously obtained values to the equations and adding the control to the first equation, we get:
 
@@ -591,7 +596,7 @@ Applying previously obtained values to the equations and adding the control to t
 .. math::
    :label: invpend2
    
-   (J_c + \frac{m_2L^2}{4})\ddot{\theta}+\frac{m_2gL\cos(\theta)}{2} + k_2\dot{\theta}L = 0 . 
+   \left(J_c + \frac{m_2L^2}{4}\right)\ddot{\theta}+\frac{m_2gL\cos(\theta)}{2} + k_2\dot{\theta}L = 0 . 
 
 
 After linerarization in the neighbourhood of :math:`\frac{\pi}{2}` we have :math:`\cos(\theta) \approx \frac{\pi}{2} - \theta`.
@@ -608,7 +613,7 @@ Defining :math:`x_1 = x, x_2=\dot{x}_1, x_3 = \theta` and :math:`x_4=\dot{x}_3`,
    0 & 1 & 0 & 0\\
    0 & \frac{-k_1}{m_1} & 0 & 0\\
    0 & 0 & 0 & 1\\
-   0 & 0 & \frac{m_2Lg}{2(J_c+\frac{m_2L^2}{4})} & -\frac{k_2L}{J_c+\frac{m_2L^2}{4}}\end{array}\right]
+   0 & 0 & \frac{m_2Lg}{2\left(J_c+\frac{m_2L^2}{4}\right)} & -\frac{k_2L}{J_c+\frac{m_2L^2}{4}}\end{array}\right]
    \left[\begin{array}{c}
    x_1 \\
    x_2 \\
@@ -623,15 +628,6 @@ Defining :math:`x_1 = x, x_2=\dot{x}_1, x_3 = \theta` and :math:`x_4=\dot{x}_3`,
    0 \\
    -\frac{m_2Lg\pi}{4J_c+m_2L^2} \end{array}\right].   
 
-Consider some moment of time :math:`t_1` and final position :math:`x_1(t_1) = x_1, x_2(t_1) = 0, x_3(t_1) = \frac{\pi}{2}, x_4(t_1) = 0`. It is required to calculate the backward reachability sets (tube) for the linearized system :eq:`invpendls` emanating from the given final position and project it onto :math:`(x_1, x_3)` subspace. It is also required to identify whether it’s possible to reach the final position from a given initial position :math:`x_1(t_1) = x_1, x_2(t_1) = v_1, x_3(t_1) = \theta_1, x_4(t_1) = \omega_1` using some admissible control function.
-
-.. raw:: html
-
-	<h2>References</h2>
-
-.. [SUN2003] L.Muñoz, X.Sun, R.Horowitz, and L.Alvarez. 2003. Traffic Density
-   Estimation with the Cell Transmission Model. In *Proceedings of the
-   American Control Conference*, 3750–3755. Denver, Colorado, USA.
-      
+Consider some moment of time :math:`t_1` and final position :math:`x_1(t_1) = x^1_1, x_2(t_1) = 0, x_3(t_1) = \frac{\pi}{2}, x_4(t_1) = 0`. It is required to calculate the backward reachability sets (tube) for the linearized system :eq:`invpendls` emanating from the given final position and project it onto :math:`(x_1, x_3)` subspace. It is also required to identify whether it’s possible to reach the final position from a given initial position :math:`x_1(t_0) = x^0_1, x_2(t_0) = v_1, x_3(t_0) = \theta_1, x_4(t_0) = \omega_1` using some admissible control function.