
Approximating the integral by trape zoidal integration and the derivative by a backward difference
equation leads to the equivalent digital transfer function (Jacquot 1981)
D(z) ¼
c
1
z
2
þ c
2
z þ c
3
z
2
z
(5:122)
where
c
1
¼ K
P
þ
K
I
T
2
þ
K
D
T
, c
2
¼
K
I
T
2
K
P
2K
D
T
, c
3
¼
K
D
T
(5:123)
The continuous-time process model is (see Section 2.7)
C
dT
dt
þ
1
R
T ¼
1
R
T
0
þ Q
H
(5:124)
where
C is the thermal capacitance of the interior space (objects and volume of air assumed to be at the
same temperature)
R is the effective thermal resistanc e of the material separating the inside and outside of the
chamber
The heater output is
Q
H
(t) ¼
v
2
(t)
R
e
(5:125)
where R
e
is the electrical resistance of the heater ...