Question

(1)Solve the following problems for the given inverted pendulum system shown in the figure and the equation of motion: $ml^2\ddot{\theta} = mgl \sin \theta + u$ (a) Obtain the input torque $u$ to attain the balance (equilibrium) at $\theta = 30$ deg. (b) Derive a linearized state-space equation for the small deviation of the state variables $\tilde{x}$ and the input $\tilde{u}$ from the equilibrium at $\theta = 30$ deg. $\tilde{x} = \begin{bmatrix} \tilde{x}_1 \\ \tilde{x}_2 \end{bmatrix} = \begin{bmatrix} \tilde{\theta} \\ \dot{\tilde{\theta}} \end{bmatrix}$

          (1)Solve the following problems for the given inverted pendulum
system shown in the figure and the equation of motion:
$ml^2\ddot{\theta} = mgl \sin \theta + u$
(a) Obtain the input torque $u$ to attain the
balance (equilibrium) at $\theta = 30$ deg.
(b) Derive a linearized state-space equation for
the small deviation of the state variables $\tilde{x}$ and
the input $\tilde{u}$ from the equilibrium at $\theta = 30$ deg.
$\tilde{x} = \begin{bmatrix} \tilde{x}_1 \\ \tilde{x}_2 \end{bmatrix} = \begin{bmatrix} \tilde{\theta} \\ \dot{\tilde{\theta}} \end{bmatrix}$

(1)Solve the following problems for the given inverted pendulum
system shown in the figure and the equation of motion:
ml^2θ̈ = mgl sinθ + u
(a) Obtain the input torque u to attain the
balance (equilibrium) at θ = 30 deg.
(b) Derive a linearized state-space equation for
the small deviation of the state variables x̃ and
the input ũ from the equilibrium at θ = 30 deg.
x̃ =
< b m a t r i x >
=
< b m a t r i x >

Added by Joseph C.

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