Page 183 - Mathematical Techniques of Fractional Order Systems

Fractional Order Error Models With Parameter Constraints Chapter | 6 171

C α T
D e 1 ðtÞ 5 A 1 e 1 ðtÞ 1 b 1 φ ðtÞω 1 ðtÞ
1 ð6:26Þ
T
ðtÞ 5 k 1 h e 1 ðtÞ;
1 ðt 0 Þ 5 e m 1 0
e m 1 e m 1
C α T
D e 2 ðtÞ 5 A 2 e 2 ðtÞ 1 b 2 φ ðtÞω 2 ðtÞ
2 ð6:27Þ
T
ðtÞ 5 k 2 h e 2 ðtÞ;
2 ðt 0 Þ 5 e m 2 0
e m 2 e m 2
where A 1 ; A 2 Aℝ n 3 n are matrices whose eigenvalues have negative real
n
parts, b 1 ; b 2 ; h 1 ; h 2 Aℝ and the triplets ðA 1 ; b 1 ; h 1 Þ and ðA 2 ; b 2 ; h 2 Þ satisfy the
Kalman Yakubovich Popov lemma, i.e., positive definite symmetric matri-
ces P 1 ; P 2 ; Q 1 ; Q 2 Aℝ n 3 n exist such that
T
A P 1 1 P 1 A 1 52 Q 1
1
P 1 b 1 5 h 1
ð6:28Þ
T
A P 2 1 P 2 A 2 52 Q 2
2
P 2 b 2 5 h 2 :
1 n
The error vectors e 1 ; e 2 :ℝ -ℝ are not accessible in this fractional EM,
1
:ℝ -ℝ are assumed to be accessible.
but only the output errors e m 1 ; e m 2
k 1 ; k 2 are unknown constants whose signs are assumed to be known (usually
the high frequency gains of the plants to be controlled/identified),
1 m
φ ; φ :ℝ -ℝ are the parameter errors, defined as
2
1

φ ðtÞ 5 θ 1 ðtÞ 2 θ ðtÞ
1
1
ð6:29Þ
φ ðtÞ 5 θ 2 ðtÞ 2 θ ðtÞ
2
2

1

with θ 1 ; θ 2 :ℝ -ℝ m the estimated parameters and θ ; θ Aℝ m the real
2
1
1
unknown parameters. On the other hand, ω 1 ; ω 2 :ℝ -ℝ m are vectors of
available signals and αAð0; 1Š.

Let’s assume that the unknown parameters θ ; θ ; k 1 , and k 2 are not inde-
1 2
pendent but related through the following linear matrix relationship
1 1
R 1 θ 1 R 2 θ 5 W ð6:30Þ
1 2
k 1 k 2
n
where R 1 ; R 2 Aℝ n 3 n are known matrices and WAℝ is a known vector.
Using the same definition for the auxiliary error than in FOEM2 with
^ ^
1
parameter constraints (6.14), where k 1 ; k 2 :ℝ -ℝ are additional estimated
1
parameters, which in this case estimate 1 and , respectively, then additional
k 1 k 2
parameter errors can be defined as
^
φ ðtÞ 5 k 1 ðtÞ 2 1
k 1
k 1 ð6:31Þ
^
φ ðtÞ 5 k 2 ðtÞ 2 1 ;
k 2
k 2
which allows writing the auxiliary error equation like in (6.16).