COMPARATIVE ANALYSIS OF
D
AND
ID
FRACTIONAL
REGULATORS FOR THE ROBUST CONTROL OF STATIC
PLANTS IN THERMAL ENERGETICS
Bela Benova
1
1
Technical University, Sofia, Bulgaria,
oxytank@yahoo.com
,
oxytank@abv.bg
Abstract
– In this paper are given the basic
properties and features of the CRONE control in
conditions of uncertainty for the plant’s model.
One of the most popular methods for the synthesis
of non-integer-order
control systems – the
approximation by a fractio-rational recursion – is
used for the conception of fractional differentiator
and integro-differentiator. The dynamic and
robust characteristics of two control systems
including, respectively, the D and ID regulators
and the same static plant, are compared.
Conclusions are made on the advantages and
inconveniences in using these fundamental types of
fractional-order controllers.
Keywords
– Fractional differentiation, fractional
integration, CRONE control, fractio-rational
recursion, robust analysis.
1.
INTRODUCTION
The particular class of the fractional-order robust
control systems
CRONE
(
Commande Robuste
d’Ordre Non-Entier
in french) [2,3] finds its
application when uncertainty is present in the plant’s
model.
CRONE
systems realize an efficacious
counteraction to perturbations, maintaining the robust
features to satisfy the requirements of the synthesis
criterion. Using the approximation by the fractio-
rational recursion are synthesized two fundamental
fractional-order controllers – a differentiator and an
integro-differentiator. The comparative analysis of
the time, frequency and robust characteristics,
observed in simulation for a preliminary set static
plant, decides about which structure performs better
control.
2.
BASIC NOTIONS IN THE ROBUST
ANALYSIS
The robust analysis [2,3], uses the following
expressions for the additive (1) and multiplicative (2)
perturbations, the system’s sensitivity (3) and
complementary sensitivity (4).
(
)
(
)
(
)(
)
(
)
()
ω
ξ
a
a
a
a
j
j
j
G
j
G
j
G
j
A
A
A
≤
=
−
=
∆
⇔
*
(1)
() ()
(
)
(
)
m
m
m
a
m
j
j
j
G
j
G
j
G
j
G
j
j
G
j
A
A
A
A
≤
=
−
=
=
⇔
*
*
*
(2)
η
j
j
G
j
R
j
e
−
=
+
=
1
1
1
(3)
(
)
(
)
j
e
j
G
j
R
j
G
j
R
j
−
=
+
=
1
1
(4)
For
the
preliminary
defined
multitude
[ ]
[]
*
G
,
G
∈
Π
, if (5) is satisfied, the system is
characterised by the robust stability, the robust
performance is obtained for (6) and if both
expressions are validated together (7), the system
possesses both robust characteristics.
(
)
∀
<
−
,
1
m
A
(5)
1
y
e
o
m
<
+
A
(6)
(
)
∀
<
+
∀
<
−
,
1
;
,
1
o
m
m
y
e
A
A
(7)
3.
SYNTHESIS OF NON-INTEGER-ORDER
CONTROL SYSTEMS: METHOD OF THE
POLYNOMIAL FRACTIO-RATIONAL
RECURSION
The large number of methods for the synthesis of
robust systems can be classified as follows: QFT
(Quantitative Feedback Theory), RMM (Robust
Model Matching), FDM (Frequency Domain
Methods) and a combination of the last two - RMM –
FDM. Knowing that the physical realization of a non-
integer
controller
is
impossible,
different
approximations of its mathematical description were
developed in the class of FDM methods, like that of
R.Agarwal using the Mittag – Leffler functions [4],