Page 186 - Mathematical Techniques of Fractional Order Systems

174 Mathematical Techniques of Fractional Order Systems

It will be assumed that the unknown parameter vectors θ , θ , k 1 , and k 2 ,
1 2
satisfy relationship (6.13) for known matrices and vector

1 1 12 8
R 1 5 R 2 5 W 5
2 0 03 19
It will be simulated the two FOEM2 using noncoupled fractional order
adaptive laws (NCFOAL) (6.8) and compare it with the case using coupled
fractional adaptive laws (CFOAL) (6.17).
In these simulations, the following initial conditions are used for the dif-
ferent variables in the FOEM2.
T T T T
e 1 ð0Þ 5 5 21 θ 1 ð0Þ 5 0 2 e 2 ð0Þ 5 1 0 θ 2 ð0Þ 5 1 0
For those cases where CFOAL are implemented, the following additional
initial conditions are used for the estimated parameters
^
^
k 1 ð0Þ 5 0 k 2 ð0Þ 5 0
In all the simulations, unity adaptive gains are used for all the AL.
The fractional operators were implemented using the NID block included
in the Ninteger toolbox for Matlab (Vale ´rio and Da Costa, 2004). The
Oustaloup numerical approximation (Oustaloup, 1991) was used for the frac-
tional operator in the NID block, where it is included as the Crone
approximation.

6.5.2 Simulation Results: Ideal Conditions
Simulations under ideal conditions will be presented first together with two
input (information) vectors; one constant and one time-varying. Let consider
the case when the input vectors ω 1 ðtÞ and ω 2 ðtÞ are given by
T T
ω 1 tðÞ 5 1 2 ω 2 ðtÞ 5 1 sinðtÞ
A simulation time of T 5 1000 s will be used, although for Oe 1 ðtÞO only
the first 50 s are shown in the figures.
Using the parameters given previously, the FOEM2 are simulated in both
cases, using NCFOAL (6.8) and using CFOAL (6.17).
Fig. 6.1 shows the evolution of Oe 1 ðtÞO and Oe 2 ðtÞO for different values of
α. For every α used, the corresponding results using NCFOAL and CFOAL
have been plotted in the same graph, for comparison purposes.
In Fig. 6.1A, the time axis has been zoomed for the first 50 seconds in
the case of Oe 1 tðÞO, for the sake of clarity. It can be seen from Fig. 6.1 that
Oe 1 tðÞO converges to zero for all α, and the convergence speed is not dramat-
ically different in the four cases presented. However, it can be seen that the