Page 178 - Mathematical Techniques of Fractional Order Systems

166 Mathematical Techniques of Fractional Order Systems

adaptive scheme (6.7), (6.8) remain bounded and that the mean value of the
2
squared norm of the output error (OetðÞO ) converges asymptotically to zero.
An interesting problem arises when the true unknown parameters of two
adaptive systems represented by FOEM2 like (6.7), (6.8), are related through
a known linear relationship. In that case, both adaptive schemes can be trea-
ted separately, using AL as (6.8) for every system. However, one can wonder
if the inclusion of the additional knowledge contained in the linear relation-
ship in the AL could improve their behaviors. This is precisely the idea
behind the FOEM2 with parameter constraints, which will be introduced and
thoroughly analyzed in the next subsection.

6.3.2 FOEM2 With Parameter Constraints
Two FOEM2 will be consider in what follows where their true unknown
parameters satisfy a linear matrix relationship. Let consider two FOEM2,
whose output error equations are defined as follows
C α T
D e 1 ðtÞ 5 A 1 e 1 ðtÞ 1 k 1 b 1 φ ðtÞω 1 ðtÞ; ; ð6:9Þ
1 e 1 ðt 0 Þ 5 e 1 0
C α T
D e 2 ðtÞ 5 A 2 e 2 ðtÞ 1 k 2 b 2 φ ðtÞω 2 ðtÞ; e 2 ðt 0 Þ 5 e 2 0 ; ð6:10Þ
2
where A 1 ; A 2 Aℝ n 3 n are matrices whose eigenvalues have negative real
parts, i.e., positive definite symmetric matrices 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
T ð6:11Þ
A P 2 1 P 2 A 2 52 Q 2 :
2
1
Vectors e 1 ; e 2 :ℝ -ℝ n correspond to the output errors, which are
assumed to be accessible. k 1 ; k 2 are unknown constants whose signs are
assumed to be known (usually the high frequency gains of the plants to be
1
n
controlled/identified), b 1 ; b 2 Aℝ , φ ; φ :ℝ -ℝ m are the parameter errors,
1 2
defined as

φ ðtÞ 5 θ 1 ðtÞ 2 θ ðtÞ
1
1
ð6:12Þ
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 but

2
1
1
unknown parameters. On the other hand, ω 1 ; ω 2 :ℝ -ℝ m are information
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

k 1 R 1 θ 1 k 2 R 2 θ 5 W ð6:13Þ
1 2
n
where R 1 ; R 2 Aℝ n 3 n are known constant matrices and WAℝ is a known
constant vector.