Design of Digital Controllers in the Presence of Random Disturbances

Design of Digital Controllers in the Presence of
Random Disturbances
A large number of systems are subject to disturbances of a random nature. In the
first part of this chapter models suitable for the representation of these stochastic
disturbances (the ARMAX models) will be presented and their properties analyzed.
Three design methods will be presented: 1. minimum variance tracking and
regulation (which minimizes the mean-square difference between the reference and
the controlled variable); 2. the approximation of the minimum variance tracking
and regulation by means of the pole placement (for the case of systems with
unstable zeros); 3. generalized minimum variance tracking and regulation.
4.1 Models for Random Disturbances
4.1.1 Description of the Disturbances
First consider the basic deterministic disturbances, namely the Dirac pulse, the step
and the ramp, represented in Figure 4.1.

170 Digital Control Systems

Figure 4.2. Deterministic disturbance models

Note that the step and the ramp may be described as resulting from the passing

of the Dirac pulse through a filter, as indicated in Figure 4.2. The corresponding
filters are known as “disturbance models”. Any deterministic disturbance may be
obtained by passing a Dirac pulse through a “disturbance model” (filter) having an
appropriate structure. Knowledge of the “disturbance model” is equivalent to
knowledge of the disturbance.
What follows is an attempt to extend this concept of the “disturbance model” to
the description of random disturbances.
By random or stochastic disturbances, is meant those disturbances, which
cannot be described in a deterministic way, given as they are not reproducible.
To provide an example of a random (stochastic) process, one can consider the
evolution of the controlled output of a plant in regulation on a significant horizon
(one day) and during several tests (several days). This is illustrated in Figure 4.3.