Page 638 - Mathematical Techniques of Fractional Order Systems

Enhanced Fractional Order Chapter | 20 609

From the robust compensator u a and the fuzzy-based adaptive laws, given
in Eqs. (20.36) (20.38), V ðÞ in Eq. (20.42) can be rewritten as
q ðÞ
t
1 1
t
q ðÞ T T T T
V ðÞ 52 e Qe 2 2 e PBB e 1 e PBw 1
2 2ρ
T
0 1 0 1
1 1 1 1 1
T
T
T
2
T
52 e Qe 2 @ B Pe2ρw 1 A @ B Pe 2 ρw 1 A 1 ρ w w 1
1
2 2 ρ ρ 2
1 1
T
T
2
#2 e Qe 1 ρ w w 1
1
2 2
ð20:43Þ
Integrating (20.43) from t 5 0to t 5 T, we have
T
1 ð 1
T
2
T
VTðÞ 2 V 0ðÞ #2 e Qedt 1 ρ w w 1 dt ð20:44Þ
2 2 1
0
Since VTðÞ $ 0, Eq. (20.35) can be rewritten as follows:
ð T ð T
T
T
T
T
e Qedt # e 0ðÞPe 0ðÞ 1 θ 0ðÞθ 0ðÞ 1 ρ 2 w w 1 dt
1 ð20:45Þ
0 0
Therefore, the robust H N tracking performance can be achieved. The
proof is completed.
The diagram of the proposed control is given in Fig. 20.1.
FIGURE 20.1 Global block
diagram of the proposed
FRAF-VSC controller.