3) (25 points) The equation of motion of a simple pendulum is given by
$ml\ddot{\theta} = -mglsin\theta - kl\dot{\theta} + T$,
where $l$ denotes the length of the rod, $m$ denotes the mass of the bob, $\theta$ is the angle subtended by the rod
and the vertical axis through the pivot point, and $g$ is the acceleration due to the gravity, $k$ is the positive
friction coefficient, and $T$ is the torque applied at the pivot point.
(i) (6 points) Setting $x_1 = \theta$ and $x_2 = \dot{\theta}$ as the state variables, $u = T$ as the input, and $y = \theta$ as the
output, obtain the state-space model.
(ii) (2 points) Find the equilibrium points for $u = 0$.
(iii) (5 points) At $x_1 = \pi$, $x_2 = 0$, and $u = 0$, linearize the system and obtain the linear state-space
model.
(iv) (2 points) Find the transfer function of the linearized model (You can use MATLAB to go from the
linear state-space model to the transfer function).
(v) (3 points) Determine whether the linearized system is BIBO stable. Explain your answer.
(vi) (2 points) For the rest of this problem, let $m = 0.1$ kg, $k = 0.1$ kg/s, $l = 0.5$ m, and $g = 9.81$ m/s$^2$.
Draw the root locus of the transfer function. (You can use MATLAB to draw the root locus).
(vii) (5 points) Does there exist a proportional controller ($K > 0$) stabilizing the linearized system?
Explain your answer.
