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ACTA IMEKO
ISSN: 2221-870X
April 2017, Volume 6, Number 1, 13-19
ACTA IMEKO | www.imeko.org April 2017 | Volume 6 | Number 1 | 13
Hybrid backward simulator for determining causal heater
state with resolution improvement of measured temperature
data through model conformation
Yukio Hiranaka, Shinichi Miura and Toshihiro Taketa
Yamagata University, Jonan 4-3-16, Yonezawa, 992-8510 Yamagata, Japan
Section: RESEARCH PAPER
Keywords: backward simulation; temperature measurement; inverse problem; precision improvement; LSB noise reduction
Citation: Yukio Hiranaka, Shinichi Miura and Toshihiro Taketa, Hybrid backward simulator for determining causal heater state with resolution improvement
of measured temperature data through model conformation, Acta IMEKO, vol. 6, no. 1, article 3, April 2017, identifier: IMEKO-ACTA-06 (2017)-01-03
Section Editor: Paul Regtien, The Netherlands
Received May 28, 2016; In final form March 14, 2017; Published April 2017
Copyright: © 2017 IMEKO. This is an open-access article distributed under the terms of the Creative Commons Attribution 3.0 License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited
Funding: This work was supported by JSPS KAKENHI Grant Number 255400006
Corresponding author: Yukio Hiranaka, e-mail: zioi@yz.yamagata-u.ac.jp
1. INTRODUCTION
To estimate heater operation from the data of room
temperature change is a typical inverse problem [1], [2]. And it
is a kind of ill-conditioned problem because a slight error in the
data would extremely disturb the results [3]. However, such a
problem is common for measurement and system diagnostics,
and an important part of measurement.
If the relation between input and output is linear, it is a
problem of deconvolution and there are many studies including
super-resolution [4]. Subtractive deconvolution [5] may be
applied if available data are impulsive. Unfortunately,
temperature change is a long trailing phenomenon and is very
sensitive to measurement resolution and error [3]. Our idea to
tackle this is to consider measurement resolution inevitable and
to treat it explicitly by defining a range for each input or output
signal.
There are two ways to search deconvolution results for
range signal models. One is a try and modify method to test
some signal within a certain divided range and to judge whether
it matches the given output data. It is a forward simulation
because it requires many trials, regardless of whether a
convergence method is used or not. The other is an inference
method starting from the given output data to the input in
backward direction. It is a normal method for problem solving,
but it is not always possible to find a good solution. However,
we may perform a backward simulation in a similar manner to
the forward simulation, starting with a range divided output
signal, if we can make a backward processing model.
There are backward simulation applications in many fields
such as process scheduling [6], initial position estimation of
physical objects [7], and software debugging [8].
ABSTRACT
We are developing a backward simulator, which determines the unknown system input from the system output by using a system
model. However, its processing time would increase enormously if the simulation model requires the multiple case branching, which is
typical for backward simulations. In some target applications, we can use forward simulation processing in the backward data flow
with significant reduction of processing time. This paper shows an example of such application to determine system input of heater
operation from measured data of room temperature. Although the resolution of measurement restricts the performance of the
simulation result, we also used the model to improve the resolution of measured data and show its effect to simulation. Furthermore,
we show the result of reduction of noise caused by quantizing LSB jitters.
ACTA IMEKO | www.imeko.org April 2017 | Volume 6 | Number 1 | 14
There exist many difficulties in creating a backward system
model. A typical backward simulation requires case branching
when it has multiple possible conditions on its backward trace.
However, we have experiences to cope with them, and have
shown that some physical restriction may effectively reduce the
number of case branches [9], [10]. We have been focusing our
attention on utilizing temporal model and strict facts such as
nonnegative property and causality that any current value must
not affect past values. Under such conditions, we can perform a
backward simulation effectively.
In this paper, we show two methods for efficiently
performing backward simulation and for effectively improving
measurement resolution. The first method is for creating a
backward simulator by incorporating a forward simulation
model. It extremely suppresses case-branching processing in the
example of this paper. The second method is for improving
resolution of the given measured data by applying a strict model
of the system.
Hereinafter, we describe the concept of backward simulation
and target simulation model in Section 1, hybrid
implementation of backward integration loop in Section 2,
measured temperature data and estimation of model parameters
in Section 3, simulation results for error free temperature data
in Section 4, effect of quantization error and countermeasure to
it in Section 5, simulation results for real measurement data in
Section 6, discussion in Section 7, and conclusion in Section 8.
2. BACKWARD SIMULATION AND MODELS
Here, we explain the concept of backward simulation with
an example application of inference of heater activity. Figure 1
shows the structure of our forward simulation model for room
temperature change caused by an electrical heater. Black dots
denote branching points, circles with plus sign denote
summation points, and arrows denote the direction of physical
information flow. A block with z-1 means one sampling time
delay, which is a member of the integration loops. We adopted
two integration loops because the room temperature increases
even after the time the heater is turned off. The first integration
loop corresponds to the neighbor of the heater that
accumulates heat locally, and the second loop corresponds to
the whole room heat reservoir. The parameters A, B are decay
factors due to heat transfer, C is the specific heat of the room
which converts heat to temperature.
Figure 1 can be expressed by the following equation,
(0 )1 ,1 ( )1
2 1 2
w Aq t tiq
Aq t ti
q Aq Bq
− ≤ <
=
− ≤
= −
(1)
where 1q and 2q are accumulated heat in the first loop and the
second loop, t is time, and the heater of w (W) is on for 0 to
ti . We can obtain 2q as
{ }
{ }
(1 ) (1 ) / ( ) (0 )
2
( 1) ( 1) / ( ). ( )
At Btw B e A e B B A t ti
q
At At Bt Bti iw B e e A e e B B A t ti
− −− + − − ≤ <
=
− − − − − − ≤
(2)
Figure 2 shows the implemented software objects and
connection diagram for the simulation of Figure 1. Each object
has a corresponding function, “heater” for heating for a
specified time duration, “wsum” for summing heat with wattage
input port, “csum” for summing heat, “br” for branching with
the same output values, “dly” for one sampling time delay, “co”
for coefficient multiplier, and “temp” for temperature
recording. These objects are coded in Scala classes, which use a
Java virtual machine, and dly1 and dly2 are instance objects of
the class dly, for example. All the objects have independent
GUI windows which accept local settings and display status and
parameters of each object.
Data flow on the connection links consist of time and
values. As an example, heating power 800 W from “heater” at
its output port “o” at the time of 10.0 s is shown in a XML
style UCF (universal communication format) message [9], [10]
heatero10.0800.0,
where sim is the simulation controller which redirects this
message from “o” port of “heater” to “i” port of “sum1” as
specified in the simulator’s connection table [10]. The tag
indicates the source of the message. The source tag is nested in
this case to indicate the port “o” of “heater.”.
In the backward simulation (Figure 3), UCF messages flow
in the backward direction indicated by dashed lines with
reversed arrows. The node “temp” is the starting object which
sends the time sequenced temperature in the reversed order,
one pair of time and temperature data in each UCF message.
The temperature in the backward simulation is expressed by a
value range which expresses the minimum and the maximum
temperature as “0.0, 10.0”, which means that the temperature is
in the range from 0.0 inclusive to 10.0 exclusive [10].
By narrowing the range, we can control the resolution of the
simulation. Simulation parameter “ndiv” set by the starting
block “temp” specifies the number of divisions. As an example,
when the whole range is from 0.0 to 10.0 and ndiv is equal to
10, the value 4.5 is expressed in the UCF message as “4.0, 5.0”
as the divided range width is 1.0.
3. HYBRID SIMULATOR AND IMPLEMENTATION
Two important features of the simulator are time
synchronization and hybrid backward simulation. Figure 4(a)
i
i
i i i
i
i i i
f
heater
wsum
br1
co1
1-A
dly1
A
1z−
dly2
B
1z−
csum
br2
co2
1/C temp
o o
o
o o1 o o o1
o
o2 o2 f
Figure 2. Forward simulation objects and connections corresponding to
Figure 1.
heater 1-A
1
C
temp
A 1z− B 1z−
Figure 1. Forward simulation model of room temperature.
heater 1-A
1
C
temp
A 1z− B 1z−
Figure 3. Backward simulation model corresponding to Figure 1.
ACTA IMEKO | www.imeko.org April 2017 | Volume 6 | Number 1 | 15
simplifies a forward integration loop in Figure 1. The
summation node must gather the same time data from the two
incoming links, the input x(i) and the feedback f(i)=y(i-1) to
form the output y(i)=y(i-1)+x(i).
A fully synchronized simulator will do the work naturally.
However, we intend to perform our simulation in a distributed
processing environment, and we prefer asynchronous
processing as long as it is possible. Then, the summation node
is designed to have an input record table which keeps time and
value pairs received at the left input port (triangle arrowed port
in Figure 1 up to Figure 13). The data message from the
feedback port (from the delay node) triggers the summation
process by picking up the data of the same time from the input
record table.
Figure 4(b) shows the backward version of Figure 4(a). We
need to calculate two backward outputs from a single backward
input at the summing point, satisfying y(i)=x(i)+f(i). The
computation intensive solution is to simulate all the pairs of x(i)
and f(i), matching the equation. A practical solution is to
perform the simulation for a finite number of divided range
pairs, e.g. {x,f} pairs of {0.0 to 2.5, 2.5 to 5.0} and {2.5 to 5.0,
0.0 to 2.5} to match the output of 4.5 [10]. A case branching
mechanism is needed for such processing.
However, if we apply another solution which uses a forward
simulation object in the backward simulation as in Figure 5, we
can avoid the expansion of processing time caused by the case
branching. In Figure 5, we can formulate the process as
x(i)=y(i)–f(i) and f(i)=y(i+1). The branch node sends received
backward data in time reverse order through the two links to
the sum node and to the delay node. The same input record
table used in Figure 4 can be utilized to keep y(i) required by
the reverse delayed feedback signal of f(i)=y(i+1).
We have to describe the detailed processing of the sum
node. In the backward simulation, data flows are expressed as a
range (the minimum and the maximum). Then, the backward
calculation must handle the range information. If the range
from the right is (a, b), which means that the minimum value is
a and the maximum value is b, and the range from the bottom
is (c, d), the output through the left port should be calculated as
(a-d, b-c) to cover the broadest value range. However, x(i) must
be positive or zero as it expresses heat. If a-d is less than zero, it
must be substituted by zero. Furthermore, if b-c is less than
zero, the case simulated is not a feasible one.
Figure 6 shows the resulting practical hybrid backward
simulator, and Figure 7 shows implemented objects and
connections. The simulation objects in Figure 7 are the same
objects as in Figure 2, with backward processing capability
except dly. The objects “co1” and “co2” divide their incoming
backward data by their coefficients, “br1” and “br2” pass
through their incoming backward data to their two backward
outputs. The simulator in Figure 7 needs three consecutive
backward inputs to start as the first data stops at csum and the
second data stops at wsum because there will be no matching
time data coming from the other backward port.
We describe here the mechanism of model mismatch
detection in detail. Sum objects (wsum and csum) in Figure 7
have two arriving inputs in the backward simulation. We note
the backward input from the right of csum at the time sample
of t as ( )v t , the other backward input can be expressed as
B ( 1)v t − because dly2 node delays the backward flow signal for
one sample time and multiplied by a constant coefficient B. The
backward output ( )u t from csum can be expressed as
equation (3) and must be positive or zero because it expresses
heat.
( ) ( ) ( 1) 0.u t v t Bv t= − − ≥ (3)
The backward output from wsum can be expressed as the
following equation,
( ) ( 1) ( ) ( 1) { (t 1) (t 2)}
1 1
( ) (t 2) ( ) (t 1)
0.1
u t Au t v t Bv t A v Bv
A A
v t ABv A B v
A
=
− − − − − − − −
− −
+ − − + −
= ≥−
(4)
If the both inequalities are not satisfied for the maximum
value of the range at any time point, the backward simulation is
failed and the starting condition must not occur for the
simulation model.
4. REAL TEMPERATURE CHANGE AND MODEL SIMULATION
At first, we verified the correctness of the simulator. Figure
8 shows the result of a room temperature measurement when
the infrared heater (800 W) on the floor was turned on for the
duration from 0 s to 180 s in a tiny room of 3.6 m3. The sensor
is a Sensirion’s SHT71, which has 0.01 degree resolution, placed
near the heater at 1 m high in the room.
Figure 8 also shows a simulated temperature change using
the parameters A (0.095/10s), B (=0.9938/10s) and C (24200
cal/deg). To determine these parameters, we assume that the
heat loss rate of the room (B) is less than that of the heater
appliance (A). Then the temperature decline after the
x(i) y(i)
1z−
f(i)=y(i-1)
(a)
x(i) y(i)
1z−
f(i)
(b)
Figure 4. Forward integration loop (a) and backward integration loop (b).
heater 1-A
1
C
temp
A 1z− B 1z−
Figure 6. Hybrid backward simulation model.
i
i
i i
i
i i i
f
o o
o
o o1 o o o1
o
o2 o2 f
heater
wsum
br1
co1
1-A
dly1
A
1z−
dly2
B
1z−
csum
br2
co2
1/C temp
i
Figure 7. Hybrid backward simulation objects and their ports.
x(i) y(i+1), y(i)
1z−
f(i)=y(i+1)
(a-d,b-c) (a,b)
(c,d)
Figure 5. Hybrid backward integration loop.
ACTA IMEKO | www.imeko.org April 2017 | Volume 6 | Number 1 | 16
temperature peak is estimated by the following equation,
log . ( )2q Bt const t ti= − + ≤ . (5)
We define t p as the peak temperature time and estimate A
by the following equation derived from equation (2) ,
( ) ( 1) / ( 1).p i iA B t At Bte e e− = − − (6)
The heat capacity parameter C is estimated as the
magnification factor for fitting the measured data with the
simulated temperature data which was calculated by using the
simulator’s forward function with the estimated parameters A
and B. Those parameters should be fine-tuned to closely fit the
measured data. However, the rear part of the curve in Figure 8
cannot be fit by our model. We may need the third heat
accumulation loop for representing wall temperature change
and heat release to the environment. As will be shown in
Section 6, the above approximation was almost enough because
large measured values around the peak are matched.
Figures 9 and 10 are the results of the backward simulation
when the simulated temperature change in Figure 8 was fed
backwardly from “temp” node in Figure 7. The width of the
resultant ranges are shown as the difference between min and
max values in those figures. The ndiv parameter was set to
1,000 and 10,000 for Figures 9 and 10, respectively. The larger
ndiv is, the closer the result is to the actual heater operation
(800 W for 0-180 s). As we calculated with a float resolution for
the temperature data, we can successfully increase ndiv up to
the desired resolution to narrow down the min-max difference.
5. EFFECT OF MEASUREMENT RESOLUTION
We may not obtain such a good result as in Figure 10 for
usual resolution data. Practically, a good resolution of
temperature measurement may be 0.01 °C. If we throw away
digits smaller than 0.01 °C from the simulated data in Figure 8,
the backward simulation will stop because of a model mismatch
for ndiv larger than 108, which corresponds nearly to 0.01 °C
resolution as the maximum temperature change is 1.2 °C
(Figure 8). Figure 11 shows the backward result for ndiv 108.
If we intend to obtain better results by increasing ndiv, we
have to improve the resolution of the backward temperature
data. Figure 12 illustrates our method to improve the
resolution, where the value div1 is original resolution and value
div2 is half of div1. Circular points indicate original A/D
truncated values for the resolution of div1. If the resolution is
Figure 8. Measured and simulated temperature change caused by heating
for 180 s from the start.
Figure 11. Backward simulation result for ndiv=108 for the resolution
limited data.
div2
div1
1 2 3 4 5 6
Figure 12. Resolution increase needs to raise values at some points.
Figure 9. Backward model simulation result for ndiv=1,000.
Figure 10. Backward model simulation result for ndiv=10,000.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 1000 2000 3000 4000t
em
pe
ra
tu
re
c
ha
ng
e(
de
g)
time(s)
measured simulated
0
1000
2000
3000
4000
5000
6000
0 1000 2000 3000 4000
he
at
er
(W
)
time(s)
min max
0
200
400
600
800
1000
1200
1400
0 1000 2000 3000 4000
he
at
er
(W
)
time(s)
min max
0
200
400
600
800
1000
0 1000 2000 3000 4000
he
at
er
(W
)
time(s)
min max
ACTA IMEKO | www.imeko.org April 2017 | Volume 6 | Number 1 | 17
improved to half of the original resolution, it is natural to raise
the values at the times of 3 and 5 (indicated by triangle points).
So, we wrote a shell script to raise the value when the backward
simulation is stopped by detecting a model mismatch.
As shown in Figure 13, if the backward simulation detects a
negative value and stops when calculating (3) or (4), we raise the
value by div2 at the corresponding time t of v(t) in the case of
(3) and u(t) in the case of (4). The process is repeated to get the
whole sequence of backward input data pass the model match
test, which means until we get a valid backward result. To
improve the resolution further, the repetition is to be done for a
new resolution value. There may be a case in which the model
match test was passed even if the data modification was not
fully done as in Figure 12. In such cases, further repetition for
resolution improvement needs to add more than one resolution
value and requires a longer processing time afterward.
Figure 14 shows the result for ndiv 10,000 after repeating
resolution improvement. Value ranges other than the duration
of heating converge to zero. Although values for the duration
of heating do not converge to the correct value of 800 W, they
clearly indicate heater activity and the average for the duration
of 0 s to 180 s is 780 W. It shows that our method of data
modification is to suppress the negative value points in (3) and
(4), and not to have a resolution improvement effect at positive
value points. This means that we will modify data when we
detect steeper temperature falls, while we infer heating when we
detect steeper temperature rises.
6. REAL DATA AND BACKWARD SIMULATION
We show the result of the backward simulator applied to the
measured data in Figure 8. The backward simulator could only
output a model conforming result up to ndiv=50 (Figure 15). It
shows a large gap between possible minimum and maximum
values. Still, the real heating power values reside within the min-
max pairs. As shown in the previous section, the min value will
go up and the max value will go down when the resolution of
temperature data improves.
The repetitive processing of temperature data modification
described in the above section improves the result, shown in
Figure 16 for ndiv 10,000. We cannot expect further
improvement even if we set ndiv larger, because the points of
min and max are very close to each other at almost all sample
times.
Checking the detail of the measured data in Figure 8, we
found that there are fluctuations like the sequence of circle
points in Figure 17, which would be the result of sampling and
truncating quantization of the solid line. Those may be LSB
(least significant bit) fluctuations caused by noises around
digital thresholds. Such setbacks are treated as the results of
Figure 15. Backward simulation result for real data from Figure 17 for
ndiv=50.
Figure 16. Backward simulation result for real data from Figure 17 for
ndiv=10,000 with resolution improvement.
inconsistency1 inconsistency2
accuracy
improvement
temperatnre
data
accuracy
improvement
i
i
i i i
i
i i
i
f
o o
o
o
o1 o o o1
o
o2 o2 f
heater
sum1
br1
co1
1-A
dly1
A
dly2
B
sum2
br2
co2
1/C
temp
Figure 13. Backward simulation with resolution improvement.
Figure 14. Backward simulation result for ndiv=10,000 with applying
resolution improvement.
1 2 3 4 5 6
div
Figure 17. Real A/D data of Figure 17 have fluctuating sample points.
0
2000
4000
6000
8000
10000
0 1000 2000 3000 4000
he
at
er
(W
)
time(s)
min max
0
1000
2000
3000
4000
5000
0 1000 2000 3000 4000
he
at
er
(W
)
time(s)
min max
0
500
1000
1500
2000
2500
0 1000 2000 3000 4000
he
at
er
(W
)
time(s)
min max
ACTA IMEKO | www.imeko.org April 2017 | Volume 6 | Number 1 | 18
some heating and the simulator making spontaneous wattage
rise in the resultant heating estimation in Figure 16.
If we suppress those fluctuations, we can eliminate
spontaneous values. It is natural to estimate the original
temperature change as the dashed line in Figure 17. Then, on
probation, we eliminated them, judging by eye with adding only
one LSB or subtracting only one LSB around fluctuation
points, leaving the points where the fluctuation is more than
one LSB. The result is shown in Figure 18, slightly smoother
than the measured data in Figure 8, especially in the tail part..
Figure 19 is the result of the backward simulation of the data
in Figure 18 for ndiv=10,000. It shows a better result than in
Figure 16. Values are large for the period from 70 to 160
seconds and almost zero after 170 seconds. The average heater
power for the duration of 0 s to 180 s is 746 W. Further
improvement with larger ndiv cannot be expected because the
min and max points almost coincide for all sample times.
7. DISCUSSION
By using the hybrid simulation model in a feedback type
simulation, no case branching is required in the backward
simulation. The backward simulator we use is for functional
evaluation and includes GUI for monitoring and manual
operations. It performs a single simulation for about 3.0 s for
100 sample data and about 3.6 s for 382 sample data. The
processing time is not proportional to the number of samples.
One cause for it may be the fact that we used multithread Java
processing on a four-core eight-thread CPU (Intel i7-3770).
Figure 20 shows the processing time for the total repetition
of model mismatch and data modification relative to ndiv.
Processing time in a case of model mismatch depends on the
time when the mismatch is detected. Average processing time
for one backward simulation is from 2.5 to 3 s. We modified
the temperature data step by step, which means that we
improved the resolution for ndiv 100, and then improved it for
200 by using the result of ndiv=100, as an example. The
coordinate of Figure 20 shows the total processing time up to
the abscissa ndiv values. Roughly, the logarithm of the
processing time is proportional to the logarithm of ndiv.
Resolution improvement using model mismatch was
successful. Although there are many possible methods for such
data modification, our method to improve by half of the former
resolution when a model mismatch is detected was shown
effective. Indeed, the curve of Figure 18 is in agreement with
the measured data in Figure 8. Our resolution improvement can
be considered as a method to search feasible temperature data
for a higher resolution.
As the estimated parameters A and B may have some errors,
they may cause effects on the results of the backward
simulation. We have to evaluate such effect, though they may
have little effect on the long range of time sequence because
those parameters were determined by a relatively long part of
the temperature data.
Although we show here only one real data simulation result,
the simulation model is simple, and the result for simulated data
is perfect. Also, the result for real measured data is almost
perfect even if our model and estimated parameters are not
perfect. So, we expect similar result would be obtained for
other measured data.
To improve estimations at periods of heating, we need
another simulation model, e.g. a model which restricts heater
wattage to one of two ranges. In such case, we need to find the
time point where the estimated heating is not feasible and to
determine which temperature data should be modified.
8. CONCLUSION
We showed that a backward simulation incorporating model
conformation and resolution improvement is very effective. It
is also shown that the backward simulation can be efficiently
performed by using our hybrid method (forward simulation
objects in backward simulation structure) by eliminating case
Figure 19. Backward simulation result of Fig.18 for ndiv=10,000 with
resolution improvement.
Figure 18. Fluctuation eliminated temperture data correspoinding to the
measure data in Figure 9.
Figure 20. Processing time including resolution improvement vs ndiv for the
cases in Section 5.
0
1000
2000
3000
4000
5000
0 1000 2000 3000 4000
he
at
er
(W
)
time(s)
min max
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 1000 2000 3000 4000
te
m
pe
ra
tu
re
(d
eg
)
time(s)
1
10
100
1000
10000
100000
100 1000 10000 100000
ac
cu
m
ul
at
ed
p
ro
ce
ss
in
g
tim
e(
m
in
)
ndiv
ACTA IMEKO | www.imeko.org April 2017 | Volume 6 | Number 1 | 19
branches. Causal input changes can be practically determined
for the simulated data and for the real measurement data. The
internal state of any system can be determined by the backward
simulation if we define the system’s model properly for
backward simulation.
In the case of data with high resolution, the backward
simulator outputs almost perfect results. For cases of limited
resolution, repetitive resolution improvement responding to
model mismatches was successfully done. We also showed that
the backward simulation with real measurement data can be
effectively done, although noise elimination was needed for
errors larger than quantising errors.
The results suggest that we can separate noise from signal by
using the backward simulation with a model conformation test.
The backward simulation offers us a new method to infer
causal inputs and internal states of various systems. We need to
study the relation among the precision of simulation output,
degree of model fitness and SNR of data for backward
processing.
ACKNOWLEDGEMENT
This work was supported by JSPS KAKENHI Grant
Number 25540006. We thank one of the reviewers who
motivated us to improve the simulation.
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