Example 2.7 in the 7th edition of the FPE textbook describes a model for an inverted cart-and-pendulum system that is similar to the Segway scooter. Let's take a more detailed look at what is neededto balance an inverted pendulum or Segway.

a. First, start with the equations of motion for this system that are given in the book (equation 2.30): \begin{array}{l}

\left(I+m_{p} l^{2}\right) \ddot{\theta}^{\prime}-m_{p} g l \theta^{\prime}=m_{p} l \ddot{x} \\

\left(m_{t}+m_{p}\right) \ddot{x}+b \dot{x}-m_{p} l \ddot{\theta}^{\prime}=u

\end{array} Derive the transfer function from the cart input U(s)to the angle O'(s).