Page 184 - Mathematical Techniques of Fractional Order Systems

172 Mathematical Techniques of Fractional Order Systems

Thus, the following CFOAL for FOEM3 with parameter constraints are
proposed

C α C α T φ ðt 0 Þ 5 φ
1 1 1 10
D φ ðtÞ 5 D θ 1 ðtÞ 52 γsgnðk 1 Þ½e m 1 ðtÞω 1 ðtÞ 1 R ξðtފ;
α ^
α
T
T
C D φ ðtÞ 5 D k 1 ðtÞ 52 γθ ðtÞR ξðtÞ; φ ðt 0 Þ 5 φ k 1 0
C
1
1
k 1
C α k 1 C α T φ ðt 0 Þ 5 φ
2 2 2 20
D φ ðtÞ 5 D θ 2 ðtÞ 52 γsgnðk 2 Þ½e m 2 ðtÞω 2 ðtÞ 1 R ξðtފ;
α ^
α
T
C D φ ðtÞ 5 D k 2 ðtÞ 52 γθ ðtÞR ξðtÞ; φ ðt 0 Þ 5 φ
T
C
k 2 2 2 k 2 k 2 0
ð6:32Þ
1
where γAℝ corresponds to the adaptive gain.
6.4.3 Stability Analysis of FOEM3 With Parameter Constraints
The stability analysis of two FOEM3 whose true and unknown parameters
are constraints through a linear matrix relationship, as described in the previ-
ous subsection 6.4.2, is presented and discussed in this subsection.
Let consider the FOEM3 with parameter constraints, given by (6.26),
(6.27), (6.16), and (6.32). The stability analysis can be done along the same
lines used for FOEM2 with parameter constraints in subsection. Thus, only
the main steps of the proofs are detailed in the following.
Let’s propose the following Lyapunov function candidate, which is posi-
tive definite and decrescent
T
T
T
2
V 5 e 1 ðtÞP 1 e 1 ðtÞ 1 jk 1 j φ ðtÞφ ðtÞ 1 1 φ ðtÞ 1 e 2 ðtÞP 2 e 2 ðtÞ
1
γ 1 γ k 1
jk 2 j T 1 2
1 φ ðtÞφ ðtÞ 1 φ ðtÞ ð6:33Þ
γ 2 2 γ 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 in a
1 k 1 2 k 2
similar way to the case of FOEM2, which together with (6.26), (6.27),
(6.32), (6.28), and (6.16) allows writing
C α T T T
D V #2 e 1 ðtÞQ 1 e 1 ðtÞ 2 e 2 ðtÞQ 2 e 2 ðtÞ 2 2ξ ðtÞξðtÞ ð6:34Þ
As can be seen from Eq. (6.34), the fractional derivative of the Lyapunov
function is negative semidefinite, so it can be concluded from Theorem 1
that the origin of the system (6.26), (6.27), (6.32) is uniformly stable. That
^
N
^
; φ ; θ 1 ; φ ; θ 2 ; φ ; k 1 ; φ ; k 2 Aℒ . Using this result in (6.16)
1 2 k 1 k 2
is, e 1 ; e 2 ; e m 1 ; e m 2
N
it can be concluded also that ξAℒ .
Also, using the same arguments than in the stability analysis of FOEM2,
it can be concluded here that, ’ε . 0, it holds that