Page 188 - Mathematical Techniques of Fractional Order Systems

176 Mathematical Techniques of Fractional Order Systems

number of initial oscillations in the transient response increases as α
increases, being the case that the lower α , the smoother the response.
In the case of Oe 2 tðÞO, it can be seen from Fig. 6.1B that it converges to
zero for every α used but in a more oscillatory way, although the conver-
gence speed is clearly different according to the FO used.
Regarding the differences between the noncoupled and the coupled case,
it can be seen from Fig. 6.1A that for the first FOEM2, the differences in the
response of both strategies are almost imperceptible in the plot, thus there is
no an evident advantage in using CFOAL.
However, for the case of the second FOEM2, it can be seen from
Fig. 6.1B that those cases using CFOAL present lower amplitudes of Oe 2 tðÞO
than the cases with NCFOAL. This, together with the fact that for a fixed α
the convergence speed is pretty much the same for both strategies, represents
an advantage for the CFOAL.
On the other hand, Fig. 6.2 shows the evolution of Oφ tðÞO and Oφ tðÞO,
2
1
for this simulation scenario. It can be seen from Fig. 6.2A that Oφ tðÞO does
1
not converge to zero for the first FOEM2. This behavior was expected, since
it has been observed from several simulation studies that when the input vec-
tor ω 1 ðtÞ does not contain enough spectral lines, the convergence of the
parameter error is not achieved. Although this is an empirical observation
from simulation results, some advances in the analytical support can be
found in Gallegos and Duarte-Mermoud (2016, 2017a,b).
It can be seen from Fig. 6.2A that Oφ tðÞO converges to a final value that
1
is lower for the case of CFOAL. This result can be useful in adaptive
schemes, since the final values of the parameters is usually related to the
magnitude of, for instance, the control signal generated in adaptive control
schemes, which means that this behavior could decrease the energy used by
the control.
In the case of Oφ tðÞO in the second FOEM2, it can be seen from
2
Fig. 6.2B that it converges to zero for both coupled and noncoupled FOAL.
It was also expected in this case, since the input vector ω 2 ðtÞ includes enough
spectral frequencies for achieving convergence (see Gallegos and Duarte-
Mermoud, 2016, 2017a,b for preliminary results). Nevertheless, it must be
mentioned that the initial magnitude of Oφ tðÞO is lower for the case using
2
CFOAL, obtaining the same conclusion stated for Oφ tðÞO.
1
6.5.3 Simulation Results: Noisy Environment

Since under real conditions processes are usually subjected to noise, it is
interesting to check how the adaptive systems behave under nonideal condi-
tions. To that extent, let’s now consider the case when the input vectors
ω 1 ðtÞ and ω 2 ðtÞ are given by

T T
ω 1 ðtÞ 5 11nðtÞ 2 ω 2 ðtÞ 5 11nðtÞ sinðtÞ