Design of Digital Controllers in the Presence of Random Disturbances

Figure 4.3. Recording of a controlled variable in regulation

By examining Figure 4.3, one observes that the evolution during one day may

be described by a deterministic function f(t), but that this function will be different
every day (f(t) is known as the “realization” of the stochastic process).
If the time for carrying out the measurement of the observed variable is fixed
(e.g. at 10 a.m.), each day (at each test) a new value will be measured (this it what
is known as a random variable). However, for all values measured every day at the
same time, statistics can be defined characterized by the mean value and the
variance of the measurements. The probabilities of the occurrence of different
values may be defined as well.
The stochastic process (partially) represented in Figure 4.3, is dependent on the
time (during a day) and on the experiment (first, second...fourth day).
More formally, a stochastic process may be described as a function f(t, ξ)
where t represents the time and belongs to the set T of real variables, and ξ
Control in the Presence of Random Disturbances 171
represents the stochastic variable (the outcome of an experiment), which belongs to
a probability space S1. For a given ξ = ξ0, the function f(t, ξ0) is a regular time
function called a realization. For fixed t = t0 the function f(t0, ξ) is a random
variable. The argument ξ is often omitted.
If the stochastic (random) process is ergodic, the statistics related to an
experiment (in our example over one day) are significant, i.e. the result obtained is
identical to that obtained from measurements taken on several experiments when
the time is maintained constant (at the same time of the day). If, in addition, the
stochastic process is gaussian, the knowledge of the mean value and of the
variance allows the probability of occurrence of a given value to be specified
(Gauss's bell – see Appendix A).
In practice, the majority of random disturbances occurring in automatic control
systems may be accurately described as a discrete-time white noise passed through
a filter. This discrete-time white noise is a random signal having an energy
uniformly distributed at all frequencies between 0 and 0.5 fs. Note that the discretetime
white noise has a physical realization, since it is a finite energy signal (the
frequency band is finite), whereas the continuous-time white noise does not
correspond to a physical reality since the energy is constant over an infinite
frequency range (infinite energy signal).
The filters that will constitute the random disturbance models will modify the
frequency spectrum of the energy distribution of the white noise in order to obtain
a distribution corresponding to the frequency distribution of energy of the various
random disturbances encountered.
The white noise has, in the random case, the same role as the Dirac pulse in the
deterministic case. It constitutes the fundamental generator signal.
The gaussian discrete-time white noise will henceforward be considered as the
generator signal. This is a sequence of independent equally distributed gaussian
random variables of zero mean value and variance σ2. This sequence will be noted
{e(t)} and will be characterized by the parameters (0, σ), in which the first term
indicates the mean value and σ is the standard deviation (square root of the
variance). A part of such a sequence is represented in Figure 4.4.