Page 179 - Mathematical Techniques of Fractional Order Systems

Fractional Order Error Models With Parameter Constraints Chapter | 6 167

The question becomes now how to adjust the parameter estimates
θ 1 ðtÞ; θ 2 ðtÞ using the information contained in (6.13). One solution that does
not use the relationship (6.13) is to adjust independently each error model,
i.e., to say, using AL with the same structure as in (6.8). This solution, how-
ever, does not consider the additional information contained in (6.13), which
could be valuable improving the behavior of the resulting adaptive system.
In order to consider this information in the problem solution, let’s define an
1
n
auxiliary error ξ:ℝ -ℝ , using the same idea as in Duarte and Narendra
(1996a). Thus it is defined ξðtÞ as
^
^
ξðtÞ 5 k 1 ðtÞR 1 θ 1 ðtÞ 1 k 2 ðtÞR 2 θ 2 ðtÞ 2 W ð6:14Þ
^ ^ 1
where k 1 ; k 2 :ℝ -ℝ are additional estimated parameters, which estimate k 1
and k 2 respectively. This auxiliary equation somehow tells how far are the
estimates from the true, unknown, and coupled values at any time instant.
Defining additional parameter errors for k 1 ; k 2 it is obtained
^
φ ðtÞ 5 k 1 ðtÞ 2 k 1
k 1 ð6:15Þ
^
φ ðtÞ 5 k 1 ðtÞ 2 k 2
k 2
1
where φ ; φ :ℝ -ℝ. Subtracting (6.13) from (6.14), and using the defini-
k 1 k 2
tion of the parameter errors (6.12), (6.15) it can be rewritten the auxiliary
error (6.14) as
ξðtÞ 5 k 1 R 1 φ ðtÞ 1 φ ðtÞR 1 θ 1 ðtÞ 1 k 2 R 2 φ ðtÞ 1 φ ðtÞR 2 θ 2 ðtÞ ð6:16Þ
1
2
k 1
k 2
Note that Eq. (6.16) can be used only for analytical purposes, because the
parameter errors φ ðtÞ, φ ðtÞ, φ ðtÞ, and φ ðtÞ cannot be implemented due to
1
2
k 1
k 2
the fact that the true parameters θ , θ , k 1 , and k 2 are unknown. For imple-
1
2
mentation purposes, the expression (6.14) should be used.
Now, coupled fractional AL are proposed of the form
α
α
T
T
C
C D φ ðtÞ5 D θ 1 ðtÞ52γsgnðk 1 Þ½e ðtÞP 1 b 1 ω 1 ðtÞ1R ξðtފ; φ ðt 0 Þ5φ
1 1 1 1 1 0
α
α ^
C D φ ðtÞ5 D k 1 ðtÞ52γθ 1 ðtÞR ξðtÞ; φ ðt 0 Þ5φ
T
C
1
D φ ðtÞ5 D θ 2 ðtÞ52γsgnðk 2 Þ½e ðtÞP 2 b 2 ω 2 ðtÞ1R ξðtފ; φ ðt 0 Þ5φ 2 0
C α k 1 C α T T k 1 k1 0
2
2
2
2
α ^
α
T
C D φ ðtÞ5 D k 2 ðtÞ52γθ 2 ðtÞR ξðtÞ; φ ðt 0 Þ5φ
C
k 2 2 k 2 k2 0
ð6:17Þ
1
where γAℝ corresponds to a constant adaptive gain. Nevertheless, adaptive
gains γ can be chosen all different for each parameter adjustment in (6.17).
For simplicity in this study all adaptive gains will be chosen constant and
equal. Furthermore, as mentioned in Section 6.3.1, it is also possible to use
time-varying adaptive gains for parameter’s adjustment in (6.17) and this
extension is being currently being investigated.
As can be seen, FOAL (6.17) include the additional information given by
the linear constraint (6.13), through the inclusion of the auxiliary error ξ tðÞ.