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Biosystems Analysis and Optimization 59

2.4.1 Excitation Experiments
To obtain good estimates for the model parameters one needs experi-
mental data that contains good information about the dynamics.
Because we want to estimate the parameters of an input–output
model, both the input and the output signals should have a signal-to-
noise ratio as high as possible with respect to the dynamic relation
between the input signal and the output signal. This can be obtained
by applying an input signal to the system that has sufficient energy to
excite the dynamics of interest. Therefore, this type of experiment is
called an excitation experiment.
When we are mostly interested in modeling the frequency domain
characteristics of the system, we will try to obtain a good empirical esti-
mate of the Bode plot in the frequency region of interest. Because the
bode plot is a plot of the amplification and phase shift by the system as a
function of frequency, we could apply sine excitations at different fre-
quencies of interest and measure the response of the system to these
frequencies in terms of amplification and phase shift. Applying different
sines one after another would, however, take quite some measurement
time because we would have to measure several periods per sine to
obtain a good estimation of the amplitude and phase in the output signal.
Therefore, it is more efficient to excite the system with a multisine, which
is a summation of sine waves of different frequencies within a certain
spectrum. The sine waves have also been shifted in phase to obtain a
periodic signal with a small crest factor resulting in high signal-to-noise
ratios (Pintelon and Schoukens 2001). Obviously, the output signals will
also be a multisine, but one where the amplitudes and phases have been
changed. This multisine can then be split into its sine components by
means of a Fourier transformation. When we have no idea about system
dynamics, a multisine with a flat spectrum (equal amplitude for all fre-
quency components) will be used. However, if we have a good idea
about the expected shape of the Bode plot, we might apply a multisine
with a different spectrum to obtain output signals with a nearly flat spec-
trum and thus have a better signal-to-noise ratio. Figure 2.14 is an exam-
ple of a multisine signal that is given with a flat frequency spectrum.
When we are interested mostly in modeling the time domain
characteristics of the system (model to be used for prediction), realis-
tic input signals should be applied. If the model should be well capa-
ble to predict the response of the system to a step input, this is applied
in the excitation experiments. This type of input has the advantage
that it also excites many frequencies, but in a more realistic combina-
tion than the multisine. This is especially important for nonlinear sys-
tems that, differently from linear systems, exhibit dynamic behavior
dependent on the input signal. Multiple upward and downward step
inputs can be combined into an efficient excitation signal (e.g., block
wave). Another commonly used input signal for time domain analy-
sis is a ramp input, which can be combined into a sawtooth signal.
Figure 2.15 shows some typical time domain excitation signals.