In general, a fast-slow dynamic system is defined as a system that has state variables that operate on two different time scales. A prime example of such systems is the cartpole problem. This system is imaged below in Figure 1 with the state variables labeled. The position and velocity of the cart denoted by x and the angle and angular velocity of the pole relative to the vertical axis denoted by 𝜽. In the standardized layout of this system, force is applied exclusively to the mass in the ±x direction, denoted in the figure by F.
Figure 1. Cart-pole System Diagram. The fast and slow dynamics, corresponding to the pole and cart respectively, can be tracked through parameters 𝜽 and x, representing the angle of the pole from vertical and the position of the cart. Force is applied to the cart in the horizontal direction.
Given the physical differences between the pole and cart, each of these elements operate on different time scales; this can be understood by considering how quickly each would respond to the applied force. The cart has a much greater mass and will therefore require more time for it to exhibit displacement than the pole would under the same force magnitude. Thus, the angle and angular velocity of the pole can be understood as the fast dynamics while the position and velocity of the mass can be viewed as the slow dynamic. The timescales for each of these components can be coupled or decoupled based on the velocity at which the cart is moving; as the cart reaches higher velocities, its timescale approaches similar proportions to the poles. For this system however, input is directly applied to the slow dynamic. This limits this system to be representative of only a specific class of fast slow dynamic systems.