IEEE Systems, Man and Cybernetics Magazine - January 2023 - 63

inside the container Tin
is the output signal y(t) from the
SN NSN 1
+T1
Lip
plant,
respectively, the plant is expressed from (8) and (9)
as follows:
P| =+ + hc
pc
Z
[
\
]
]
]
]
CRc
in
yt Tin
() =
2C ut Cc
.
2 ++ k
rp
dTin
dt
2
c
a1
2
R
R
()
w
c
ST
ut CR
TT
in - CR Tin
-
1
cc
()
cp
Next, we discuss operators N and D .1The
designed operators N and D 1N
:()(
)
()
Z
[
\
]
]
]
]
cp
yt Tin
Dw t :()()()
-12p
c
ST
=- Cc
pc
ut C
r
- 2 ut
CRc
in
dTin
dt
2
2 ++ k
a1
R
R
w
c
-
CR
T
cc
c
(13)
Operators N and D 1=
in
- CR Tin
1
cc
=- +- -
=
wt CR TT CR Tc
11
hc
cc
(14)
(15)
where N is stable, and D is stable and invertible.
Next, the operators S and R are designed to satisfy the
Bezout identity, shown as
(( ))
SN wt RD wt Iwt+= (16)
(( )) (( ))
where I is the identity operator. By designing the stable
operators S and R to satisfy the Bezout identify, the plant
becomes the right coprime factorization and BIBO stability.
The designed operators S and R are shown as
Sb tB CRc
:( )( );
Ru t =- ++rp
-1 :( )
ST
pc cm
ST
pc
rp
in
cc
=- -- +
++ -
11
1
dy t
2
2
dt
()
2
()
a
CR yt CR
TT
cp
rB
2Ce t
p
c
hc
-
()
R
R
w
c
CR
T
k
cc
c
dy t
dt
D
()
(17)
(18)
where B is a design parameter and takes a value between
zero and one.
When the uncertainty NT
is taken into account, the
control system is robustly stable when it satisfies the equation
expressed as
()=+ TS
ut Ke t
Pt()
t
P + ∆P
∆N
u ∈ U
r ∈Y+
et ∈Y
C1
−
b ∈U
S
Figure 3. The control system for the calorimeter.
January 2023 IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE 63
ri ∈ U+ e ∈U
−
R−1
D−1
w ∈W
N
+
+
y ∈Y
[( ( ))]Ke t
a
# edt xx
t
()
(24)
dt
dT
() .
(19)
Passivity
Passivity is an essential element in nonlinear control,
including system stabilization and adaptive control system
design. Passivity is satisfied to guarantee the stability of
the adaptive control system [17]. First, the definition of passivity
is explained. Consider the following nonlinear system
with u(t) as the input and y(t) as the output of the system:
are designed to satisfy the right factorization PN .D 1
-
are expressed as
dt
dT
(
wt fu t
yt gw t
=
=
() (( ))
() (( ))
.
(20)
The system shown in (20) is said to be passive when it satisfies
the following equation using the storage function
(( )),
Vw t which is a quasi-positive constant function:
(( )) (( ))
Vw tVwu yd() ()
-
0 # xx x
#
where () ()uy
t
(21)
xx is called the supply rate. Equation (21)
implies that the amount of energy increase in the system is
less than the total of the energy given from outside sources.
When the accumulation function is differentiable, (21)
can be rewritten as
o (( )) # () ().
Vw tu ty t
lowing equation:
(( ))
(22)
The calorimeter is passive because it satisfies the folVw
tw tr ty tw ty t
o ==i#() () () () ().
(23)
Adaptive Control System Design
Next, we design a compensator to track the internal temperature
to a target value. In this article, adaptive control
theory is used to respond to changes in the measurement
object's heat capacity. The adaptive control system C1
used for the tracking compensator is designed as
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