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164 Mathematical Techniques of Fractional Order Systems
6.2.3 Additional Tools for the Analysis of FOS
The following lemma will be used in this chapter to prove boundedness of
the signals in the fractional order error models (FOEM) analyzed here.
n
Lemma 1: (Duarte-Mermoud et al., 2015). Let xðtÞAℝ be a vector of differ-
entiable functions. Then, for all t . t 0 , the following relationship holds
(Duarte-Mermoud et al, 2015, Lemma 4)
1
T
a
C D a x Px ðtÞ # x ðtÞP C D xðtÞ; ð6:4Þ
T
2 t 0 t0
where αAð0; 1 and PAℝ n 3 n is a constant, square, symmetric and positive
definite matrix.
Particular cases of relation (6.4) (Duarte-Mermoud et al., 2015, Lemma
4) were proposed in Aguila-Camacho et al. (2014, Lemma 1) and Alikhanov
(2010, Lemma 1).
In order to analyze the evolution of the output error in the FOEM treated
in this chapter, the following lemma will be used.
1
Lemma 2: (Aguila-Camacho and Duarte-Mermoud, 2016) L. et xðUÞ:ℝ -ℝ
be a bounded nonnegative function. According to Aguila-Camacho and
Duarte-Mermoud (2016, Lemma 4), if there exists some αAð0; 1 such that
1 ð t xðτÞ
12α dτ , M; ’t $ t 0 ; with MAð0; NÞ ð6:5Þ
ΓðαÞ ðt2τÞ
t 0
then
" t #
Ð xðτÞdτ
α2ε t 0
lim t 5 0; ’ε . 0 ð6:6Þ
t-N t
6.3 ANALYSIS OF FRACTIONAL ORDER ERROR MODEL 2
WITH PARAMETER CONSTRAINTS
From stability and performance viewpoints, many adaptive systems (identifi-
cation and/or adaptive control schemes) can be described by a couple of dif-
ferential equations; one describing the evolution of output error and the other
that of the parameter error. This pair of equations are called Error Model
(EM). Thus, all the conclusions obtained from the study of this couple of
equations regarding stability and performance, are applicable to any adaptive
scheme that can be put in this form. This is precisely the main attraction of
EM, since it gives a general framework to analyze adaptive systems.
