Another system which captures fast slow dynamical behavior with input applied to the fast dynamic is the acrobot system. This system is shown below with relevant state variables labeled.
Figure 1. Acrobot System Diagram. The top hinge allows angular movement, its position is held constant. Input is applied as torque to the second hinge. Since a large change in the second link results in a small change to the first link, the first link can be considered the slow dynamic while the second link is the fast dynamic. Each can be tracked through parameters 𝜽2 and 𝜽1 representing the angle of the first link and the relative angle of the second link.
As seen in Figure 1 above, θ1 represents the angle the first link makes from the vertical axis and θ2 represents the angle that the second link is offset from the first link. Torque, denoted 𝛕, is applied to the hinge between the first and second link. Finally, the first link’s top hinge is bolted down with the ability to rotate, but unable to be displaced. As with the cartpole system, examining the physical interplay between these two links can reveal some behavioral differences between the two links. As torque is applied to the hinge, the second link is affected much more quickly and experiences significantly more displacement. For first link to experience any substantial displacement, it requires much more time and that the second link to take on a variety of different positions. As such, this system can be identified as another fast slow dynamic system. The timescales for each of these dynamics can be coupled or decoupled based on the relative mass of each link; as the first link takes on more mass, it requires more time for the fast dynamic to affect the slow dynamic and thus can be interpreted as a decoupling of the time scales. Inversely, as the first link grows in mass to match the first link, the timescales for the fast and slow dynamics begin to overlap. It is important to note that the input to this system is applied directly to the fast dynamic as applying torque to the join is mathematically equivalent to applying a force to the second link.