232
en(y-y") + e„! (S
' -
1) - E,;(y-y°)
Chapter 6
d
( v —
v°)
dt
(32)
d(p'-p°)
dt
Epjy-y0
) - D (p’- p ’°)
; p ’
= p/s°
In the following we shall repeat the perturbation experiment of eq (22) to (25). At constant
5
’ = 1 y is
perturbed from its steady state value
and the transient of
y
and of
p'
back to their steady state values is
followed. The solution of the two last equations of (32) for initial values y (t=0) = y(0), p’(0) = p’° is :
y(t) - v(0) = - (y(0) - y°) ( 1
- exp((EM-
6
,
2
) 0)
P
(t
) -p
cn(v( 0)-v°)
----------------—- (exp(-Dt) - exp((En - el2)
0
)
e,2
- e u
-D
(33)
When the first equation, multiplied by a = £l
2
/(Ei
2
-e
1
i-Z>) is added to the second equation the terms in
exp((En - e
12
)
t)
cancel and
a (y(0)
-y) + (p' -p'°) =
a (y(0) -y°) ( 1 - exp(-
D
t))
(34)
Except for the factor a the right hand side of (34) is a known function/(r) of time. Except for a the left
hand side of (34) is also a known function of time. Consequently a is determined by regression of
-(p '(Q)-p')
(y(
0
)-y)-f(t)
(35)
using the experimentally determined values of the deviation variables
(y
- y(
0
)) and
(p'~
p'°) at a set of
/-values. Knowing a for two experimental runs with different values of
D
permits the elasticities en and
E
12
to be determined by simple algebraic calculations. Finally the control coefficient
C'
is determined
using eq (17). Note that the same steady state value
5
° can be maintained for the two different
D
values
by a simultaneous change of
to keep
constant in the two experiments.
In silico
experiments show that this variant of the Delgado and Liao transient method is much more
robust with respect to the influence of experimental errors than the original method. A series of 10
experiments were run (/ values the same as in Fig. 6.10 and 6.11) with respectively
D
= 1 and
D =
0.1
(same unit for (tim e1) as used in
k,
and
k2).
C / was determined to 0.911 ± 0.015 when the data was
overlaid with random noise
10
% relative to the mean, and this is much better than the result obtained
previously with only
2
% relative noise.
In this numerical experiment we have kept
s
at
throughout the experiment. If s is allowed to deviate
from s°, i.e. if we make a simulation of the real pulse experiment, then the elasticity
£01
(= 0.11778 in
(14)) can also be found, but we must of course now measure all three deviation variables as functions of
time during the transient. If
e0, =
0
then measurement of
s' -
1
contains no information of value for