![](bg7.png)
The energy distribution based on the complex
atomic dictionary in Figure 4 is better localized in
time and frequency than the other. Also its resolu-
tion is much higher, and its high-energy deterministic
components can be much better separated from low-
energy stochastic components.
As we have announced, we are not only interested
in separating deterministic and stochastic components
in control signals, but also in separating their con-
stituents, i.e. periodic, subharmonic, chaotic and tran-
sient for deterministic nonlinear time-varying com-
ponents versus non-stationary and stationary for sto-
chastic components. The non-stationary and deter-
ministic chaotic vibrations present in the servo error
signal have narrow high-frequency support and are
difficult to be detected using Wigner distribution. By
applying the adaptive decomposition algorithm de-
scribed in Section 2.2.2, we are also able to identify
such narrow-band frequency characteristics, as can be
seen from the ‘banana’ shape of the dark patterns in
Figure 4, which are much more expressive than the
black patterns in Figure 2.
In Figure 2 the local cross-terms around t =
{0, 0.1, 0.2, 0.3} are not eliminated, while in Figure
4 they are not present anymore.
The energy plot in Figure 4 does not contain cross-
terms and any numerically undesired effects, as in
Figure 2.
Additionally, our multi-dictionary approach leads
to a clearer identification of time-varying non-linear
and stochastic servo error components and therefore,
to an improved adaptive ILC design.
We further observe that an overall smaller band-
width than in Figure 2 is obtained (not higher than
about 400 [Hz] in the beginning of the acceleration
profile and 100 [Hz] around other jerk moments).
Therefore, the ILC needs to learn only around these
jerk moments up to smaller frequencies than those
found when Wigner distribution was applied for the
computation of the bandwidth profile.
This means that the stochastic effects will not be
amplified unnecessarily (high bandwidth means good
tracking performance and noise amplification) while
all existent deterministic effects will be learned. Also,
unlike in Figure 2, we do not obtain any increased
bandwidth because of cross terms: the bandwidth of
the Q-filter needs to be increased just around the jerk
moments while, in between, a small bandwidth can be
maintained.
Also, as seen in Figure 4, the bandwidth profile has
a very good smooth approximation. Therefore, fast
switching between the cut-off frequency of Q-filters
that corresponds to different time instances is avoided
and the stability of the switched ILC system is not an
issue anymore.
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