Hybrid systems comprise continuous and discrete-event subsystems that interact. The continuous parts, unlike the discrete-event parts, cannot be simulated in their original form. Instead, continuous models are approximated with discrete-event models amenable to computer simulation. Any number of approximating systems can be used, each answering in a different way three fundamental questions:

In any finite interval of time, the continuous system traverses an infinity of states, but estimates can be computed for only a finite number of them (i.e., the discrete system must be legitimate); which points are picked?

To calculate these points requires knowledge of the trajectory between them; what is assumed?

The system interacts with its environment; how are inputs and outputs, continuous and discrete, handled?

For the moment, consider only questions 1 and 2 in relation to a single, ordinary differential equation

This continuous model can be approximated by (1) discretizing time into points separated by intervals of duration h and (2) assuming that x follows a line in those intervals. This yields the discrete-time system

which estimates x at times t0+kh for k = 0, 1, 2,…. Indeed, this is Euler’s method for the numerical solution of an ordinary differential ...

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