Design of Digital Controllers in the Presence of Random Disturbances

Figure 4.4. Discrete-time white noise sequence

Control in the Presence of Random Disturbances 173

Since E{e(t)} = const. and E{e(t) e(t-i)} is only a function of i, the gaussian

discrete-time white noise sequence has the properties of a weakly stationary
stochastic process (in fact it is a real stationary stochastic process since the
properties of the sequence {e(t)} are the same as those of the sequence {e(t+τ)}).
The knowledge of the covariances R(i) for a weakly stationary stochastic process
makes it possible to compute the energy distribution in the frequency domain,
known as the spectral density function.





Figure 4.5. Normalized autocorrelations of the white noise

This is given by (discrete Fourier transform of the covariance function)

Since in the case of the discrete-time white noise all the R(i) = 0 for i ≠ 0, it results
that the spectral density of the discrete-time white noise is constant and equal to

The spectral density of the white noise is represented in Figure 4.6. A uniform
energy distribution between 0 and 0.5 fs is observed.
Figure 4.6. Spectral density of the discrete-time white noise


4.1.2 Models of Random Disturbances

As it has already been mentioned in Section 4.1, different types of random

disturbances, whose spectral density can be approximated by a rational function of
the frequency, can be considered as resulting from the filtering of a white noise
through a shaping filter. Several types of processes thus obtained will be examined.
“Moving Average” Process (MA)
Consider, for example, the process

which corresponds to the filtering of a white noise through a filter (1+c1 q-1), as
shown in Figure 4.7a.





Figure 4.7a,b. Generation of a “Moving Average” random process: a first order; b general
case
The mean value of y(t) is 3

The variance of the process y(t) is

since the third term is zero (see properties of the white noise, Equations 4.1.3 and

4.1.6).
Control in the Presence of Random Disturbances 175
From Equation 4.1.7, one obtains by shifting