Question

(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.

          (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.

(b) (25 points) The equations describing the cart-pole system are
(mc + mp)ẍ + mp l θ̈cosθ - mp l θ̇^2 sinθ = fx
mp l ẍcosθ + mp l^2 θ̈ + mp g l sinθ = 0
(i) Find the equilibrium positions for θ. Linearize the above system about θ = π.
(hint: ignore higher order terms)
(ii) Given a suspension system:
m z̈2 + b ż2 + (k1 + k2)z2 = b ż1 + k1 z1
m z̈1 + b ż1 + k1 z1 = k1 z2 + b ż2 + F
Write the equation in state-space form, i.e., ẋ = Ax + Bu. The input is u = F.
Write the output equation y = Cx, if the output is y = z2.
Your answer should describe the state-vector x, and the A, B, C matrices.

Added by Melissa B.

Question

Please give Ace some feedback