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near their self-selected cadence for five minutes.
The second ergometer test consisted of four
sprints of 6 s duration each and an incremental ramp
test. Two sprints were carried out before and two after
the ramp test to obtain the subjects’ maximal power
output and
˙
V O
2
profiles in a recovered and a fatigued
state.
In the third test subjects had to complete a vari-
able step protocol. The steps varied in load and dura-
tion and alternated between low and moderate or se-
vere intensity. The linearly in- or decreasing intensity
between the steps was also varied in time. The load
profile is illustrated in Figure 1.
For the final ”synthetic hill climb test” the er-
gometer was controlled by our simulator software in
Dahmen et al. (2011). The load was defined by the
mathematical model by Martin et al. (1998) to simu-
late the resistance on a realistic track. The gradient of
that track and the subjects’ body weight were the ma-
jor determinants of the load. While holding the same
cadence as before, the subjects were able to choose
their exercise intensity by gear shifting. (On the steep-
est section most subjects were not able to hold the ca-
dence even in the lowest gear.)
Data Preprocessing
In order to validate and compare the models, data se-
ries of time-stamped values of produced power and
resulting breath-by-breath oxygen consumption are
required for exercise intensities ranging from moder-
ate to severe. These time series from ergometer labo-
ratory experiments are typically very noisy, have dif-
ferent sampling rates and the samples may be irregu-
larly spaced.
Therefore, a combined smoothing and resampling
operator has to be applied. In this study we have
used the standard Gaussian smoothing filter with ker-
nel (σ
√
2π)
−1
exp(−0.5t/σ
2
) and σ = 20 s for respi-
ratory gas, heart rate and power measurements.
Dynamic Model
We have extended the dynamic model from Ar-
tiga Gonzalez et al. (2015) described above with
two more parameters. With these two additional pa-
rameters, a much smaller average root-mean-square
modeling error was obtained and also the predictive
power of the model was improved (details to be pub-
lished elsewhere). For a better comparison between
the dynamic model and Hammerstein-Wiener mod-
els, we also applied this modified dynamic model for
˙
VCO
2
and heart rate modeling and prediction based
on power.
3 HAMMERSTEIN-WIENER
MODELS
The dynamical model for
˙
V O
2
under variable work
rate (Artiga Gonzalez et al., 2015) described in Sec-
tion 2 is based on physiological evidence collected in
many years of research. Thus, in addition to the ap-
plication for modeling and prediction, the estimated
model parameters can be used as indicators for per-
formance capabilities of athletes or enhance the com-
prehension of physiological processes. For instance,
a deeper analysis of the second differential equation
might lead to a better understanding of the so called
slow component.
Black box models like Hammerstein-Wiener mod-
els do not offer the same understanding as physio-
logical models have, but they bring other advantages.
Detached from physiological evidence they are more
flexible and can adjust better to data and therefore,
may deliver better fitting results.
Though not derived by principles of physiology, it
is still important to select the right model type and
model settings to obtain good results. For this re-
search MATLAB
R
was used. The System Identifica-
tion Toolbox
TM
offers a large selection of models. Dif-
ferent linear (ARX, ARMAX, State-Space) and non-
linear (ARX, Hammerstein-Wiener) models from that
toolbox have been tested on selected data sets with
the System Identification App. Best results have been
achieved with State-Space and Hammerstein-Wiener
models. In a direct comparison Hammerstein-Wiener
models have shown the best modeling results.
This outcome coincides with the knowledge that
there is a strong linear relationship between power
and
˙
V O
2
(fast component) and a smaller nonlinear
relation (slow component), because Hammerstein-
Wiener models consist of nonlinear and linear ele-
ments. This holds also for the relationship between
power and
˙
VCO
2
or heart rate.
Figure 2: Block diagram of Hammerstein-Wiener model.
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