The pendulum shown in the figure may be controlled by manipulating the acceleration of

the cart carrying it$.$ The rotating pendulum has a moment of inertia $I=m{L}^2$ about its pivot.

a$)$ Derive the equations of motion for the pendulum relating the pendulum angle $$ to the

cart acceleration $u=v^{}.$ Show that the resulting equation of motion is

$L{}^{}+v^{}cos+$gsin$=0$

b$)$ Use the equation of motion in $($a$)$ to find input $u(t),0t{t}_f,$ that moves the pendulum

from $=0$ to $=-\frac{}{3}$ in $t_f$ seconds according to $(t)=\frac{}{3t_f^3}{t}^2(2t-3t_f).$

c$)$ Formulate a state variable model $($SVM$)$ with $u$ as input and $v,,$ and ${}^{},$ as state

variables.

d$)$ Develop a Matlab Simulink model based on the SVM in $($a$).$ Provide the Simulink

diagram with your homework.

e$)$ Simulate the Simulink model in $($c$)$ assuming $L=\frac{1}{2}m$ and $g=9\mathrm{.}81\frac{m}{se{c}^2}$ for $u(t)$ you

found in $($b$)$ with $t_f=3$ seconds. Did $u(t)$ do what you expected?