Page 181 - Mathematical Techniques of Fractional Order Systems

Fractional Order Error Models With Parameter Constraints Chapter | 6 169

Additional conclusions regarding the evolution on time of output errors
e 1 ðtÞ; e 2 ðtÞ and the auxiliary error ξðtÞ can be drawn from the following
analysis.
Let’s apply the FOI of order α to expression (6.21), from which it is
obtained
α T α T α T
I e Q 1 e 1 ðtÞ 1 I e Q 2 e 2 ðtÞ 1 2I ξ ξ ðtÞ # Vðt 0 Þ 2 VðtÞ ð6:22Þ
t 0 1 t 0 2 t 0
Since it was proved that all the signals of the FOAS remain bounded,
from expression (6.18) it can be concluded that V remains bounded ’t $ t 0 .
Thus, from (6.22) it can be concluded that the three FOI in the left-hand side
of (6.22) remain bounded ’t $ t 0 .
This result, together with the fact that e 1 ðtÞ; e 2 ðtÞ and ξðtÞ are bounded
and the arguments of the FOI are nonnegative, allow using Lemma 2 to con-
clude that, ’ε . 0, it holds that
" # " #
2
2
Ð t Oe 1 ðτÞO dτ Ð t Oe 2 ðτÞO dτ
α2ε t 0 α2ε t 0
lim t 5 lim t
t-N t t-N t
" #
2
Ð t OξðτÞO dτ
α2ε t 0
5 lim t 5 0 ð6:23Þ
t-N t
Expression (6.23) implies that the mean values of the squared norm of
the output errors e 1 ðtÞ; e 2 ðtÞ and the auxiliary error ξðtÞ are oðt ε2α Þ, ’ε . 0,
which means that they converge asymptotically to zero, with a convergence
speed higher than t 2α .
6.4 ANALYSIS OF FRACTIONAL ORDER ERROR MODEL 3
WITH PARAMETER CONSTRAINTS
This section presents the same analysis performed in Section 6.3 but for two
FOEM3 whose true and unknown parameters satisfy a matrix linear
constraint.

6.4.1 Fractional Order Error Model 3

FOEM3 has the same structure as FOEM2, but the main difference is that
n
only one of the components of the error eðtÞAℝ is accessible. This situation
is very common when access to the whole pseudo state of the plant to be
controlled/identified is not available but only one component is measurable
(plant output). This makes FOEM3 applicable to a much wider class of pro-
blems than FOEM2.
In FOEM3, the evolution of the output error e 1 ðtÞAℝ has the form