The classic inverted pendulum is an example of a trivially underactuated system : it has two degrees of freedom ( one for its support's motion in the horizontal plane, and one for the angular motion of the pendulum ), but only one of them ( the cart position ) is actuated, and the other is only indirectly controlled.
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A second type of inverted pendulum is a tiltmeter for tall structures, which consists of a wire anchored to the bottom of the foundation and attached to a float in a pool of oil at the top of the structure that has devices for measuring movement of the neutral position of the float away from its original position.
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PS optimal control theory has been used in ground and flight systems The techniques have been extensively used to solve a wide range of problems such as those arising in UAV trajectory generation, missile guidance, control of robotic arms, vibration damping, lunar guidance, magnetic control, swing-up and stabilization of an inverted pendulum, orbit transfers, tether libration control, ascent guidance and quantum control.
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