Page 81 - Biosystems Engineering

62 Chapter Two

voltage applied to a hydraulic valve into the height of a slurry injec-
tor above the soil (Saeys et al. 2007). Six periods of a multisine rang-
ing from 0.02 to 4 Hz have been applied to the system at a sample
frequency of 40 Hz. Because the output signal for the first period con-
tains transient behavior of the system, this period has been removed
from the data prior to calculating the Fourier transforms.

2.4.3 Estimating Nonlinearities
The larger part of our system-analysis tools is aimed at linear sys-
tems, whereas in practice most biologically related systems exhibit at
least some nonlinear behavior. To investigate the contribution of sys-
tem nonlinearities compared to linear system behavior and determine
whether a linear model can be sufficient, we can further analyze the
response of the system to a multisine.
For instance, apply a random odd multisine to the pump setting
of a hydraulic pump and observe the response of the ground speed.
This example is further elaborated later and can be seen in Fig. 2.3. A
multisine is the sum of sines where the phase of each sine is random.
Frequencies of the sines are multiples of the given base frequency,
which is the lowest frequency contained in the multisine. For an odd
multisine, only odd multiples of the base frequency are excited.
Because the presence of harmonics is a good measure of nonlinearity,
only exciting odd multiples of the base frequency enables us to quan-
tify nonlinearities in the system. After all, the first (and most power-
ful) harmonic of an odd frequency is the double frequency, which, of
course, is even and thus not excited directly. The magnitude of the
response on even frequencies is a good measure for nonlinearity.
The response on the second harmonic is also very interesting.
Because the second harmonic is again an odd multiple, it is very hard
to distinguish the harmonic response from the excitation response. By
removing one-fourth of the odd frequencies randomly from the excita-
tion sequence, the system response on these nonexcited odd frequen-
cies becomes a good indication of the second harmonic response.
The response on the nonexcited frequencies needs to be compared to
a reference to be able to determine whether the effect is important or not.
This reference is the noise level of the frequency response. The noise level
can be estimated by applying multiple periods of the same multisine and
calculating the variation of the response over the periods.
A multisine experiment has been performed on the propulsion sys-
tem of the combine harvester with a sampling rate of 20 Hz. The length
of the period was set to 1024 points (or approximately 50 s). This means
that the base frequency is 20/1024 = 0.0195 Hz. Because the machine is
not expected to respond to frequencies much higher than 1 Hz, the mul-
tisine only excites the frequency range between 0.0195 and 2 Hz. Four
and one-half periods of the multisine are applied to the process. After the
transition phase is removed, four full periods are available for analysis.