Page 177 - Mathematical Techniques of Fractional Order Systems

Fractional Order Error Models With Parameter Constraints Chapter | 6 165

In the IO case, four EM have been completely studied (Narendra and
Annaswamy, 2005). With the introduction of fractional operators in adaptive
schemes, the generalization of these four EM to the FO case arises.
Important advances in the analysis of FOEM have been made in the last
years (Aguila-Camacho and Duarte-Mermoud, 2016), although there are
some topics currently under investigation. The so called FOEM1 with param-
eter constraints has already been thoroughly studied in Aguila-Camacho and
Duarte-Mermoud (2015,2017) and its main properties were analyzed. In this
chapter the focus is on the FOEM2 and FOEM3 with parameters constraints,
which will be stated in what follows.

6.3.1 Fractional Order Error Model 2

One of the FOEM studied in Aguila-Camacho and Duarte-Mermoud (2016)
n
is the FOEM2, where the output error eðtÞAℝ is given by the following
fractional order differential equation
C α T
D eðtÞ 5 AeðtÞ 1 kb φ ðtÞωðtÞ; eðt 0 Þ 5 e 0 ; ð6:7Þ
where AAℝ n 3 n is a matrix whose eigenvalues have negative real parts, i.e.,
positive definite symmetric matrices P; QAℝ n 3 n exist such that
1
A P 1 PA 52 Q. e:ℝ -ℝ n corresponds to the output error, which is
T
assumed to be accessible. k p is an unknown constant whose sign is assumed
to be known (usually the high frequency gain of the plant to be controlled/
identified). The sign of the high frequency gain can be also unknown, and in
this case an alternative solution using the Nussbaum gain has been proposed
for integer order error models (IOEM) (Narendra and Annaswamy, 2005)
n
and it is being currently investigated for the FOEM2 case. bAℝ ,
1 m
φ:ℝ -ℝ is the parameter error vector, defined as φðtÞ 5 θðtÞ 2 θ ðtÞ, with
1 m m
θ:ℝ -ℝ the estimated parameters and θ Aℝ the true but unknown para-
1
meters. ω:ℝ -ℝ m is a function vector of known signals usually called
information vector. The analyses in this paper will be performed for
αAð0; 1Š.
It was proved in Aguila-Camacho and Duarte-Mermoud (2016) that adap-
tive law (6.8) can be used to estimate the unknown parameters θðtÞ through
α
α
C
T
C D φðtÞ5 D θðtÞ 52 γ sgnðkÞe ðtÞPb ωðtÞ; φðt 0 Þ 5 φ ; ð6:8Þ
0
1
where γAℝ is the adaptive gain that allows to handle the parameters con-
vergence. A straightforward extension of this adaptive law is the case when
T
the adaptive gain is given by a matrix Γ 5 Γ . 0, with ΓAℝ m 3 m . Time-
varying adaptive gains ΓðtÞ satisfying certain conditions are also an option.
This approach successfully used in the IOEM case is a topic currently under
research for the FOEM case. Using adaptive law (6.8), it was proved in
Aguila-Camacho and Duarte-Mermoud (2016) that all the signals in the