System Examples

A picture is worth 1K words, or something like that. Let's examine some sample trajectories of representative systems to develop a sense of the context we'll be operating in. Please note that the illustrations are seperate, located at the end of the paper. A basic example is the following discretization of two-dimensional harmonic motion.

We'll start the system at , and assume that the various noise inputs have diagonal covariance matrices (ie, have independant components). In fact, we'll go so far as to assume that the components are of equal variance as well, so that we may describe the Random Vectors with scalar variances, say and .

1. We begin with the system unperturbed by noise of any kind. Note that the natural behavior is a gentle outward spiral (not a circle since this is only an approximation of harmonic motion).(both variances zero.)
2. Next let's introduce some error into our observations,( )
3. and into the state sequence itself.( , ) itself.

In our science-fiction example, what could this system represent? As the oil slick spreads outward from the foundering supertanker (inebriated pilot cursing, waving his fist from the railing) we might expect to see some outward radial motion from its denizens. It is not much of a stretch to imagine temperature differences and atmospheric conditions creating a rotational flow about the ship. These two together make the outward-spiralling tansfer Function behavior we see here. The various forms of noise have the usual interpretations mentioned at the outset.