In any real-time control system, there is always some amount of
external noise 
, and usually this noise is stochastic
in nature. The distribution and magnitude of this noise depends on the
working environment, and sometimes it is too difficult to prevent
the noise from happening, but we can modify the control model to
reduce the effect of such noise to an acceptable degree. This noise
can be modeled using statistical measures and some assumptions about
its nature. To deal with this noise we must assume that
it is bounded, that is, there is a constant a such that:
This maintains the property of a stable linear system known as bounded-input bounded-output (BIBO) stability.
As a simple case, assume that 
is a constant. In this
case, the steady state error can be calculated by analyzing the system
at rest (i.e., set all derivatives to zero) as follows:
or
The value of e here represents the steady state error of the system.
From the last equation, it is clear the increasing kp will
decrease the steady state error. On the other hand, there is a limit
on the value of 
to maintain the stability of the system.
Another way to reduce (and sometimes eliminate) the steady state error, is by adding an integral term to the control low. That is what is known as the (PID) Proportional, Integral, Derivative controller. By adding this term, the steady state error can be calculated as follows:
or
We assumed, however, that is a constant, thus,
, which gives:
So, the addition of this integral element can eliminate constant disturbances.