Page 182 - Mathematical Techniques of Fractional Order Systems

170 Mathematical Techniques of Fractional Order Systems

C α T
D eðtÞ 5 AeðtÞ 1 b φ ðtÞωðtÞ; eðt 0 Þ 5 e 0
ð6:24Þ
T
e m ðtÞ 5 kh eðtÞ; e m ðt 0 Þ 5 e m 0 ;
where AAℝ n 3 n is a matrix whose eigenvalues have negative real parts.
1
n
e:ℝ -ℝ corresponds to the nonaccessible pseudo state error vector and
1
e m :ℝ -ℝ is the measurable output error. k is an unknown constant with
known sign (usually the high frequency gain of the plant to be controlled/
1
n
identified); b; hAℝ ; φ:ℝ -ℝ m is the parameter error, defined as
1 m m
φðtÞ 5 θðtÞ 2 θ ðtÞ, with θ:ℝ -ℝ the estimated parameters and θ Aℝ the
1 m
real and unknown parameter vector. ω:ℝ -ℝ is a vector of available sig-
nals and αAð0; 1Š.
As might be expected, more stringent conditions have to be imposed on
the FOEM3, in order to compensate for the fact that only e m ðtÞ is accessible.
In this case, positive definite symmetric matrices P; QAℝ n 3 n must exist
T
such that A P 1 PA 52 Q and Pb 5 h, which implies that the triplet
fA; b; hg satisfies the Kalman Yakubovich Popov lemma (Narendra and
Annaswamy, 2005).
As in the case of FOEM2, when α 5 1in (6.24), the case of classic (inte-
ger) error model 3 (IOEM3) arises, which has been completely studied
(Narendra and Annaswamy, 2005). In the case when αAð0; 1Þ, it was proved
in Aguila-Camacho and Duarte-Mermoud (2016) that the following AL
α
α
C D φðtÞ 5 D θðtÞ 52 γ sgnðkÞe m ðtÞωðtÞ; φðt 0 Þ 5 φ ; ð6:25Þ
C
0
1
where γAℝ is the adaptive gain, guarantee that all the signals in the adap-
tive scheme (6.24), (6.25) remain bounded and that the mean value of the
squared norm of error eðtÞ converges asymptotically to zero. As pointed out
in Section 6.3.1 for the FOEM2, the sign of the high frequency gain need
not to be known and in this case the solution makes use of the concept of
Nussbaum gain proposed for IOEM (Narendra and Annaswamy, 2005). This
extension is currently under investigation for the FOEM3 case.
After the analysis of FOEM2 with parameter constraints in Section 6.3.2,
it is natural to wonder if coupled fractional order AL can be derived also for
two FOEM3 whose real unknown parameters are related through a linear
relationship. This question is addressed in the next subsection.

6.4.2 FOEM3 With Parameter Constraints

The problem of two FOEM3, as described in the previous subsection 6.4.1,
whose true and unknown parameters are constraints through a linear matrix
relationship is stated in this subsection.
Let consider two FOEM3, whose output error equations are defined as
follows