If we are to study filtering in stochastic control systems, a natural first question is, ``What are Stochastic Control Systems?''
A Control System is, in barest generality, a mathematical model in which the observer of the system has some input back into the behavior of the system. The Dynamics of the system are then determined by the interactions between the observer's input, called the control, and the inherent behavior of the system, labelled somewhat obliquely the transfer function. The observer may also not have direct access to the state of the system itself, but only to some image or echo thereof. That is: In control theory, we often have access not directly to the variables we wish to model or control, but to some intermediary variables which are related thereto. These intermediaries, known as the measurements or observations, allow us to analyze systems with more state variables than observation variables.
This incestuous little epistemological knot is made still more torturous when we consider the possibility that not only our measurement of the system, but our knowledge of the system itself, is imperfect. We can model these imperfections in measurement and model structure by introducing random variables to our mathematical description of the system.
A natural second question is of course, ``What is Filtering?''
To filter is to refine.
What we're trying to refine in this context is the ``quality'' of our information about the current state of the system. That is, we'll be given a time-ordered, discrete sequence of observations of the system, which are known to be in error. Our job will be to build another sequence which gives a better idea what the system is actually doing, gives a ``more accurate'' description of the system's behavior. Exactly how this is done, and how we define ``better'' will have to wait for the moment, but it is important to understand the goal which motivates the structures we will be developing in the meantime. This attempt to sift out the imperfections in our perceptions, to see through the unavoidable errors in any attempt to describe or understand, through to some clean truth lying beneath the confusing surface; this will be the break in the stark horizon, stain-glass glowing in evening gloom, toward which we point our steps. I hope the landscape we traverse enroute may have its own rewards as well.
As an example to carry throughout this paper, we may hypothesize that we are some futuristic
nanotechnology behavioral engineer, studying the migratory patterns
of micro machines which graze oil spills. Given the thin, laminar
nature of an oil slick on water, we may model the motion of a molecular machine inside it
as a trajectory
in two-dimensional euclidean space, 
. In this context, we might
define a Stochastic Control System where the transfer function
represents some linear flux in the environment (currents due to thermal
gradients, waves, whatever); the control will model the behavior of the
nano-beastie (note that this may be quite sophisticated and non-linear since we assume it known);
``Plant Noise'' can be considered effects of unknown fluid dynamics; and
Observation Noise can be the model of refraction involving the different
optical qualities of water, air, and oil in combination with
sunspots and UFOs.
We may extrapolate further upon our little science-fiction plot.
There were always concerns that the self-reproducing machines might, when released into the environment, spread beyond their intended domain and impact the surrounding ecostructures. Proponents of the scheme argued that the machines, designed to survive only in their intended niche, would be unable to spread, and that the benifits, in terms of oil-spill damage prevention, far outweighed the risks. While the verdict on that balance has yet to be known, escape the nano-beasties surely did. This presents a new challenge for our protagonists the intrepid Stochastic Control Theorists. Their entire observation and tracking system was set up for a two-dimensional state space. Now that the meta-nisms have escaped into the three-dimensional undersea world, are they beat? Never! With a speed born of desperation, the resourceful Graduate Student intern replaces all the matrix algebra routines in the system with newer algorithms which use psuedo-inverses, and so can cope with a situation such as this where the observations are of lesser dimension than the state. With a sigh of relief, they all update their latex files, and settle in to see what further crises the day may bring.