FAQ: In a difference equation, how is the state continuous even though the time is discrete?

Submitted by: Michael Reiser
Submitted on: October 8, 2003

This type of system model arises naturally in many problems. Consider for example the predator-prey from Monday's lecture. In this model the state vector (consisting of the number of rabbits and the number of foxes) is treated as real-valued (though of course thinking about non-integer animal quantities seems silly), and the rate at which we update the population occurs at a discrete time interval, one year or one months or similar.

Perhaps a better example is any sample-based system. Suppose our cruise controller uses some standard microprocessor system, that samples the instantaneous car velocity at some fixed rate. Here the sampling in our sensor generates the discreteness of the problem, but of course the quantity we sample can take on any value. We could try to write continuous time control laws for this system, but it seems foolish to do so, since we only get information about our system at discrete time steps. One simple thing we might want to do with these sampled values is smooth them so we are less likely to be effected by a sensor error. This can be done with a difference equation that implements a moving average, where the state vector keeps track of the last few measurements.