Inverted pendulum

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1 Inverted Pendulum

The inverted pendulum is a standard controls example and it is related to rocket and missile guidance, where thrust is actuated at the bottom of a tall vehicle. Many undergraduate controls courses use the inverted pendulum as their first example plant. This classic problem in dynamics and control theory is widely used as a benchmark for testing control designs.

1.1 Equations of Motion[1]

Inverted Pendulum
Inverted Pendulum Free Body Diagram

Equations of motion usually start with Newton's 2nd Law and the inverted pendulum is no different. Sum of the torques must equal 0.

LaTeX: \sum\tau=\frac{d}{dt}\left[L\right]


where
LaTeX: \tau is torque and
LaTeX: L is angular momentum.

The torques are the control torque LaTeX: \tau and the gravity (positional) based torque LaTeX: lmg\sin\left(\theta\right)

LaTeX: \tau+lmg\sin\left(\theta\right)=\frac{d}{dt}\left[L\right]


where
LaTeX: l is the length of the pendulum,
LaTeX: m is the mass at the end of the pendulum,
LaTeX: g is gravity and
LaTeX: \theta is the angle of the pendulum off vertical.

Angular torque is LaTeX: L=I\ddot{\theta}. The inertia of the rod is LaTeX: I=ml^2 so the angular torque is

LaTeX: \tau+lmg\sin\left(\theta\right)=ml^2\ddot{\theta}


Rearranging the 2 sides

LaTeX: \ddot{\theta}=\frac{g}{l}\sin\left(\theta\right)+\frac{1}{ml^2}\tau IP 1


Eqn. IP 1 is nonlinear due to the sine function. However, if we can assume that LaTeX: \theta is small (approx. 3 degrees or less) then the small angle approximation (LaTeX: \sin\left(\theta\right) \approx \theta) is valid and IP 1 becomes

LaTeX: \ddot{\theta}=\frac{g}{l}\theta+\frac{1}{ml^2}\tau IP 2



1.2 State-Space Formulation

The typical state-space formulation includes a description of the system via state variable

If we make the substitution LaTeX: x=\begin{Bmatrix}\theta \\ \dot{\theta}\end{Bmatrix} and LaTeX: u=\tau then


LaTeX: \dot{x}=\begin{bmatrix}0 & 1 \\ \frac{g}{l} & 0\end{bmatrix}x+\begin{bmatrix}0 \\ \frac{1}{ml^2}\end{bmatrix}u


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