Problem $1-$ Apply the Lagrange Method

The figure below shown an Inverted Pendulum, a very common problem in nonlinear control courses. A

mass $m$ is connected via a massless rod of length $l$ to a cart of mass $M,$ as shown in Figure $1.$ The cart

is moving along the x $-$direction shown in figure, while the rod can rotate freely around the revolute joint

at the center of the cart. A force $f(t)$ is applied to the mass $M$ along the x $-$axis and the acceleration

due to gravity acts downward.

rigure $1.$ anverteu f enuuinin.

For this problem, you are asked to do the following:

Assuming the position of the cart $x(t)$ and the angular position of the pendulum var$(t)($measured

as shown in figure$)$ as the generalized coordinates, write the expressions defining the position and

velocity of the cart and the pendulum on the $(x,y-$plane;

Write expressions for the Lagrangian function $L$ and the Rayleigh dissipation function $D$ of the

system;

Write the equations of motion for the inverted pendulum by applying the Lagrange method;

Write the equations of motion in state$-$space form, and define the state vector;

Determine the expressions for the two equilibrium conditions of the inverted pendulum, when the

external force is $f(t)=0.$ Comment on the solutions found.