(b) (25 points) The equations describing the cart-pole system are
$(m_c + m_p)\ddot{x} + m_p l \ddot{\theta} \cos\theta - m_p l \dot{\theta}^2 \sin\theta = f_x$
$m_p l \ddot{x} \cos\theta + m_p l^2 \ddot{\theta} + m_p g l \sin\theta = 0$
(i) Find the equilibrium positions for $\theta$. Linearize the above system about $\theta = \pi$.
(hint: ignore higher order terms)
(ii) Given a suspension system:
$m \ddot{z}_2 + b \dot{z}_2 + (k_1 + k_2)z_2 = b \dot{z}_1 + k_1 z_1$
$m \ddot{z}_1 + b \dot{z}_1 + k_1 z_1 = k_1 z_2 + b \dot{z}_2 + F$
Write the equation in state-space form, i.e., $\dot{x} = Ax + Bu$. The input is $u = F$.
Write the output equation $y = Cx$, if the output is $y = z_2$.
Your answer should describe the state-vector $x$, and the A, B, C matrices.
