Page 180 - Mathematical Techniques of Fractional Order Systems
P. 180
168 Mathematical Techniques of Fractional Order Systems
It is interesting to note that by making ξðtÞ 0in (6.17) the independent
adjustment of estimates is obtained (noncoupled fractional order adaptive
laws, NCFOAL). Stability analysis of the adaptive systems (6.9), (6.10)
using coupled fractional order adaptive laws (CFOAL) (6.17) will be
addressed in the next subsection.
6.3.3 Stability Analysis of FOEM2 With Parameter Constraints
Since the two adaptive systems whose output error described by (6.9), (6.10)
are now linked through the relationship (6.17), combined FOAL as stated in
(6.17) will be used. Thus it can be defined an enlarged adaptive system given
by the set of Eqs. (6.9), (6.10), (6.16), (6.17), for analysis purposes.
In order to prove the stability of the adaptive scheme (6.9), (6.10), (6.16),
and (6.17) let’s use the fractional extension of Lyapunov direct method, spe-
cifically Theorem 1, proposing the following Lyapunov function candidate,
which is positive definite and decrescent,
T
T
2
T
V 5 e ðtÞP 1 e 1 ðtÞ 1 jk 1 j φ ðtÞφ ðtÞ 1 1 φ ðtÞ 1 e ðtÞP 2 e 2 ðtÞ
γ
γ
1
2
1
1
k 1
jk 2 j 1
2
T
1 φ ðtÞφ ðtÞ 1 φ ðtÞ ð6:18Þ
γ γ 2 2 k 2
Using the results by Miller and Feldstein (1971) it can be proved that
e 1 ; φ ; φ ; e 2 ; φ , and φ are differentiable, thus Lemma 1 can be used
1 k 1 2 k 2
allowing to write
α
α
α
α
C
T
C
C D V # 2e ðtÞP 1 D e 1 ðtÞ 1 2jk 1 j φ ðtÞ D φ ðtÞ 1 2φ ðtÞ D φ ðtÞ 1
C
T
1 1 1 k 1 k 1
T C α T C α C α
1 2e ðtÞP 2 D e 2 ðtÞ 1 2jk 2 j φ ðtÞ D φ ðtÞ 1 2φ ðtÞ D φ ðtÞ
2 2 2 k 2 k 2
ð6:19Þ
Evaluating the derivative (6.19) along (6.9), (6.10), and (6.17) and group-
ing similar terms, the following expression is obtained
C α T T T T
D V # e ðtÞ½A P 1 1 P 1 A 1 e 1 ðtÞ 1 e ðtÞ½A P 2 1 P 2 A 2 e 2 ðtÞ 2
1 1 2 2
T
T
T
2 2 k 1 φ ðtÞR ξðtÞ 2 2 k 2 φ ðtÞR ξðtÞ 2 ð6:20Þ
T
1 1 2 2
T T T T
2 2 φ ðtÞ θ ðtÞR ξðtÞ 2 2 φ ðtÞ θ ðtÞR ξðtÞ
k 1 1 1 k 2 2 2
Replacing expressions (6.11) and (6.16) in (6.20) it can be rewritten that
C α T T T
D V #2 e ðtÞQ 1 e 1 ðtÞ 2 e ðtÞQ 2 e 2 ðtÞ 2 2ξ ðtÞξðtÞ ð6:21Þ
1 2
As can be seen from Eq. (6.21), the fractional derivative of the Lyapunov
function is negative semidefinite, so from Theorem 1 it can be concluded
that the origin of the system (6.9), (6.10), (6.16), (6.17) is uniformly stable.
N
^
^
That is, e 1 ; e 2 ; φ ; θ 1 ; φ ; θ 2 ; φ ; k 1 ; φ ; k 2 Aℒ . Using this result in (6.16) it
2
1
k 2
k 1
N
can be concluded also that ξAℒ .
