Page 190 - Mathematical Techniques of Fractional Order Systems

178 Mathematical Techniques of Fractional Order Systems

where nðtÞ corresponds to a random noise uniformly distributed, with maxi-
mum amplitude of 0.5, which represents 50% of the magnitude of the ideal
reference. A simulation time of T 5 1500 s will be used, although for
Oe 1 ðtÞO only the first 50 s are shown in the figures. The the evolution of
Oe 1 ðtÞO, Oe 2 ðtÞO in this noisy scenario is presented in Fig. 6.3. The evolution
of the norm of the parameter errors Oφ ðtÞO, Oφ ðtÞO, on the other hand, is
2
1
presented in Fig. 6.4.
It can be seen from Figs. 6.3 and 6.4 that the noise indeed affect the evo-
lution of the signals in the scheme. In order to characterize and have a better
insight on the behavior of the adaptive schemes when noise is present, the
following two indexes are calculated
ð T
2
ISN 1 5 Oe 1 ðτÞO dτ ð6:37Þ
t50
T
ð
2
ISN 2 5 Oe 2 ðτÞO dτ ð6:38Þ
t50
Table 6.1 shows the resulting values of indexes (6.37) and (6.38) for the
simulations, which helps appreciating the results numerically.
As can be seen from Table 6.1, for the first FOEM2 the ISN 1 is lower for
the case of using CFOAL compared with the case of using NCFOAL, for
every α used, which represents an important improvement in the system
behavior. In the case of the second FOEM2, it can be seen from Table 6.1
that for α 5 0:5, the NCFOAL behaves clearly better than the CFOAL, e.g.,
the ISN 2 is lower for the NCFOAL. However, this behavior starts changing
as α increases, and for α 5 0:7 both approaches have similar ISN 2 . For the
cases α 5 0:9 and α 5 1 the CFOAL behaves better than the NCFOAL.
Summarizing, it can be concluded that the use of CFOAL can lead to an
improvement in the adaptive systems behavior. Thus, they should be consid-
ered as a possible alternative in the design stage, wherever it is possible to
implement them.

6.6 CONCLUSION

In this chapter, it has been proved that given two fractional order EM 2 with
linear constraints on their true unknown parameters, it is possible to derive
coupled fractional AL including the information of the constraint, guarantee-
ing stability of the resulting adaptive system for αAð0; 1Š. The same conclu-
sions have been established for two fractional order EM 3, also with linear
parameter constraints. Also, it was analytically proved that the mean value
of the squared norm of the state error vector converges asymptotically to
zero in both cases.
Besides, simulation studies showed that the inclusion of the additional
information in the coupled fractional AL results in a better dynamic behavior