2.6 Stochastic Systems

A system that always responds identically to the same set of initial conditions and control input signals during multiple test runs is deterministic. If there is some variation in the system behavior from run to run even though all initial conditions and control inputs remain the same the system behavior is described as stochastic, or random. An example of stochastic behavior is an aircraft flying with a predefined sequence of control inputs. Wind gusts affect the flight path differently on each attempt, so although we model the aircraft deterministically, the wind model in this case is stochastic. If any part of a mathematical model is stochastic, we describe the model itself as stochastic.

In addition to external random disturbances such as wind gusts, there may be randomness associated with the system itself. Manufacturing tolerances may produce variations from unit to unit in parameters that affect the performance of the system. We model these random variations in the parameters of the system and its operational environment in a stochastic manner.

First, consider the case where a single number is required to specify a random effect that we wish to model. The example we will use is the misalignment of a system component due to manufacturing tolerances. The alignment is specified as an angle, a. To meet specifications, this angle must lie in the range from a _min to a _max and is equally likely to have any value in this range. Absolute limits...