The general modeling and control of systems has been an interesting topic in engineering investigation.  Control of fast slow systems can take the form of an optimization problem, seeking to minimize some cost metric in taking the system from some initial state to another desired state in the state space[1].  Figure 1 depicts a helpful geometric interpretation of this problem.

Figure 1. Geometrical Interpretation of system trajectories for reaching a desired state.  Note, y(x1,x2,u) represents the cost exhibited at various state and input combinations, x2 represents the fast dynamics, and x1 represents the slow dynamics.  Points (x10,x20) and (x1f,x2f) represent the initial and desired state of the system.  Typically, the fast dynamic must be driven to stabile state before the slow dynamics of the system can be affected as demonstrated by the green state trajectory.  This is represented by the fast dynamic, x2 needing to be minimized before affecting the slow dynamic, x1.

 

 The fast-slow dynamic interaction of the system parameters allow for an interesting set of available methods used to control the system.  Often, the control mechanism must first address the fast dynamic, getting it to some state, before it can adequately influence, directly or indirectly, the slow dynamic.  Visually, this can be interpreted in the Figure 1, where x2 represents the fast dynamic and x1 represents the slow dynamic.  The green trajectory indicates how the fast dynamic must be moved to a certain region, in this case minimized, before accomplishing substantial movement of the slow dynamic.

The cartpole and acrobot systems both exhibits fast slow dynamics and can thus be posed with control objectives that seeks to emphasize this tradeoff of slow and fast dynamics.  For the cartpole, this can be structured as a control problem seeking to keep the pole vertically balanced while the cart reaches a certain velocity. In this case, the balancing of the pole would be interpreted as stabilizing the fast dynamic and the velocity of the cart as the desired slow dynamic state.  On the other hand, the objective for the acrobot control problem focuses on moving the first link into an upright position, i.e. θ1 = 𝛑, in the smallest amount of time possible. Instead of seeking to keep the fast dynamic in a particular state, the objective of the fast dynamic was allowed to change and vary over time in an attempt to maximize the resulting effect on the slow dynamic.