IEEE Systems, Man and Cybernetics Magazine - January 2023 - 64

where gain K is the adaptive gain and is expressed as [23]
(( ))
o
Ke te t
et
tt
t
=*
ett
,
bm
bm
()
-
() -
(())
(())
(( )).
- ##
et
et
t
t
1
2
ett
-
m
In (24) and (25), a , b , and m are design parameters,
where a and b control how quickly the gain increases.
When ()
#m the gain increase stops.
The Design of a Feedback Control System
Based on Isomorphism
Next, we discuss design control systems that satisfy passivity
based on isomorphism and operator theory are discussed.
The feedback control system based on
isomorphism is shown in Figure 4 [20], [21].
Plant P and operators N and D 1are
the same as in
(13), (14), and (15). In order for the control system to satisfy
passivity, D 1u
is
designed as
:()().
u
Dw tu t
-1
=
Next, Q 1be
D
.1u
2
1
1
is
shown as
Qu t :()().
-1
=- +- 2
rp
ST
pc
ST 2
pc
rp
Finally, the operators Su and Ru
SN wt RD wt Iwt+=
u
u :( )( );
cc
1
=
Sb tB CRc
in
()
Su and Ru are shown as follows:
2
2Cc utrp
(26)
is designed for the feedback control system to
The designed operator Q 1(27)
are
designed based on
operator theory to sat isfy the Bezout ident ity
(( ))
uu (( )) (( )). The designed operators
f =
1
=- -- +
++ -
11
1
dy t
2
dt
CR yt CR
TT
cp
Ru t B et2
1
u - :()( ).
hc
-
()
a
CR
T
cc
c
D
R
R
w
c
k
dy t
dt
()
(28)
(29)
The Experiment
This section describes the experiment. The section " The
Experimental System " describes the experimental system,
The Experimental Results
The experimental results for the DUT (heater and heat
sink) heat capacity of 300 J/K are shown in Figures 6-10.
From Figure 6, it can be seen that the internal temperature
follows the target ambient temperature in about 1,700 s.
Figure 7 shows the temperatures of the radiation side and
the endothermic side of the Peltier element, and Figure 8
shows the input current. The result of the quantity of the
D˜ −1
˜
r ∈Y˜
+
−
e ∈Y
R˜ −1
u1 ∈U˜ +
−
u2 ∈U˜
Q−1
˜
u ∈U˜
D−1
w ∈W
N
P
˜
y ∈Y
mm
m
and the section " The Experimental Conditions " describes
the experimental conditions. The section " The Experimental
Results " shows the results of the experiments.
(25)
The Experimental System
The configuration of the system used for the experiment
is shown in Figure 5. Also, Table 4 shows the
details of the equipment used in the experimental system.
In the experiment, a 3.3-Ω aluminum foil heater
and a heat sink for changing the thermal capacity are
alternatively used as the measurement object to the
power conversion device. The heater is driven to a
power consumption of 15 W in the chamber. As the
internal temperature of the chamber is raised by the
heater, the inside of the chamber is cooled by flowing
electric current through the Peltier element. At this
time, the current flowing to the Peltier element is controlled
so that the internal temperature of the chamber
follows the ambient temperature. When the internal
temperature of the chamber matches the temperature
outside the chamber, the exothermic quantity of the
heater is estimated from the endothermic quantity of
the Peltier element.
The Experimental Conditions
We conducted experiments under the conditions shown in
Table 4 and calculated the amount of heat generated by the
heater (the DUT) from (12). Also, the heat capacity of the
DUT is changed by using a heat sink. The error in the measured
value is calculated from the following equation:
uQin
Qin
c -
# 100 .
(30)
˜
b ∈Y
Figure 4. The feedback control system based on isomorphism.
64 IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE January 2023
S˜
IEEE Systems, Man and Cybernetics Magazine - January 2023

Table of Contents for the Digital Edition of IEEE Systems, Man and Cybernetics Magazine - January 2023
