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In this case the influence of the mixture of the modes in the transition region becomes
negligible. The derived estimate for appropriate values of
τ
could be confirmed for the
evaporator model.
It is important to note that the trajectories of the hybrid model and the smooth ap-
proximation are nearly identical outside the transition region. Obviously, the smoothing
only extends the transition time but does not drive the system to a different region of the
state space. From these results, we can conclude that our smoothing approach is well
suited for the evaporator model.
For a more quantitative analysis of the convergence of the solutions of the relaxed
model to that of the original model, we consider in Figure 4 and 5 the average squared
deviation
x
(
hybrid
)
(
t
i
)
− x
(
smooth
)
(
t
i
)
max
[
x
(
hybrid
)
]
2
N
s
=
1
N
(12)
i
=1
for the state variables
x
=
ξ
C
(Figure 4),
T
evap
(Figure 5(a)) and
p
evap
(Figure 5(b))
calculated with the hybrid and the smooth model, respectively. For the vapor mass frac-
tion
ξ
C
the average squared deviation is found numerically to follow approximately
s
=
ατ
+
with
α
=0
.
35
and
=1
·
10
−
4
. Theoretically we expect
=0
for sufficiently well behaved functional dependencies and arbitrarily fine discretiza-
tion (
N →∞
). In this case, the dependence of the deviation of the discontinuous state
on the smoothing parameter
τ
is dominated by the finite width of the transition region.
For continuous state variables such as
T
evap
and
p
evap
the contribution of the transition
region is expected to be small and to vanish superlinearly for
τ →
0
. This is confirmed
by Figure 5.
5
Parameter Estimation and Sensitivity Analysis
A fundamental task frequently occurring in process engineering is parameter estima-
tion. Parameter estimation in general aims at extracting the best guesses of the param-
eters determining the dynamics of the system under consideration based on a series of
measurements
x
(
m
)
ij
of several state variables
x
i
,i
=1
, ..., M
at different points in time
t
j
,j
=1
, ..., N
. It is useful to combine the parameter estimation of a hybrid dynamic
system with the sensitivity analysis for at least two reasons: First, the sensitivity with
respect to the smoothing parameter is needed to predict the suitability of the smooth
model for parameter estimation. Second, the sensitivities of the measured state vari-
ables with respect to the parameters to be estimated allow to evaluate whether certain
data can be used in a specific parameter estimation problem.
Figure 6 shows the sensitivities of the state variables
ξ
C
,
T
evap
and
p
evap
with re-
spect to the smoothing parameter
τ
. The sensitivity of the state variable
ξ
C
(solid line)
quantifies the observation already stated qualitatively in Section 4 that the vapor mass
fraction is influenced by the smoothing only in the transition region, i.e.,
dξ
dτ
≈
0
out-
side the transition region. The shape of
dξ
C
dτ
is easily understood in view of the trajectory
shown in Figure 3. As
τ
increases, i.e.,
Δτ
=
τ
2
− τ
1
>
0
,thecurve
ξ
C
(
τ
2
)
lies above
that of
ξ
C
(
τ
1
)
as long as
p<p
c
and thus
Δξ
C
=
ξ
C
(
τ
2
)
− ξ
C
(
τ
1
)
is negative, whereas
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