Editing Inverted pendulum (section)

=== From Lagrange's equations ===
The equations of motion can be derived using [[Lagrangian mechanics|Lagrange's equations]].  We refer to the drawing to the right where <math>\theta(t)</math> is the angle of the pendulum of length <math>l</math> with respect to the vertical direction and the acting forces are gravity and an external force ''F'' in the x-direction. Define <math>x(t)</math> to be the position of the cart.

The kinetic energy <math>T</math> of the system is:

:<math>
T = \frac{1}{2} M v_1^2  + \frac{1}{2} m v_2^2,
</math>

where <math>v_1</math> is the velocity of the cart and <math>v_2</math> is the velocity of the point mass <math>m</math>.
<math>v_1</math> and <math>v_2</math> can be expressed in terms of x and <math>\theta</math> by writing the velocity as the first derivative of the position;
:<math>
v_1^2=\dot x^2,
</math>
:<math>
v_2^2=\left({\frac{\rm d}{{\rm d}t}}{\left(x- \ell\sin\theta\right)}\right)^2 + \left({\frac{\rm d}{{\rm d}t}}{\left( \ell\cos\theta \right)}\right)^2.
</math>
Simplifying the expression for <math>v_2</math> leads to:
:<math>
v_2^2= \dot x^2 -2 \ell \dot x \dot \theta\cos \theta + \ell^2\dot \theta^2.
</math>

The kinetic energy is now given by:
:<math>
T = \frac{1}{2} \left(M+m \right ) \dot x^2 -m \ell \dot x \dot\theta\cos\theta + \frac{1}{2} m \ell^2 \dot \theta^2.
</math>

The generalized coordinates of the system are <math>\theta</math> and <math>x</math>, each has a generalized force.
On the <math>x</math> axis, the generalized force <math>Q_x</math> can be calculated through its virtual work

:<math>
Q_x\delta x=F \delta x,\quad Q_x=F,
</math>

on the <math>\theta</math> axis, the generalized force <math>Q_\theta</math> can be also calculated through its virtual work

:<math>
Q_\theta\delta\theta=mgl\sin\theta \delta \theta,\quad Q_\theta=mgl\sin\theta.
</math>

According to the [[Lagrangian mechanics|Lagrange's equations]], the equations of motion are:
:<math>
\frac{\mathrm{d}}{\mathrm{d}t}{\partial{T}\over \partial{\dot x}} - {\partial{T}\over \partial x} = F,
</math>
:<math>
\frac{\mathrm{d}}{\mathrm{d}t}{\partial{T}\over \partial{\dot \theta}} - {\partial{T}\over \partial \theta} = mgl\sin\theta,
</math>
substituting <math>T</math> in these equations and simplifying leads to the equations that describe the motion of the inverted pendulum:

:<math>
\left ( M + m \right ) \ddot x - m \ell \ddot \theta \cos \theta + m \ell \dot \theta^2 \sin \theta = F,
</math>

:<math>
\ell \ddot \theta - g \sin \theta = \ddot x \cos \theta.
</math>

These equations are nonlinear, but since the goal of a control system would be to keep the pendulum upright, the equations can be linearized around <math>\theta \approx 0</math>.