Therefore the linearized equation of motion is:

ml$2$ $+$ c $$ $+$ mgl$($sin $0+$ cos $0$ $)=$ u$($t$)(1)$

$($a$)$ Find all equilibrium points of the non$-$linear system $($i$.$e$.$ given eq$.(1)$ and $=$ $=$ $=$ u$($t$)=0$

solve for all $0).$ Compute the transfer function G$($s$)=$ $($s$)$

U $($s$)$ for each equilibrium point and determine

if the LTI approximation of the pendulum near the equilibrium is stable or unstable.

$($b$)$ Create a non$-$linear model of the pendulum and a linear model of the pendulum $($linearized at $0=0)$

in Simulink.

$($c$)$ With an initial condition of the pendulum hanging down, $=0,$ simulate the response of both

models to a constant torque of amplitude of $100$ N$-$m applied for $0\mathrm{.}01$ seconds $($basically a discretized

approximation of the unit impulse function$).$ Show the response of both of the models using a scope.

Compare the responses of the system.

$($d$)$ Simulate the response of both models with a constant torque of amplitude of $1000$ N$-$m applied for $0\mathrm{.}01$

seconds. Explain the differences between the simulations of the two models. Is the linear approximation

of the nonlinear pendulum model more accurate for the larger input, or the smaller input?

$(40$ points$)$

$2$