Chemical Engineering Science 61 (2006) 1516 – 1527
www.elsevier.com/locate/ces
System identification of the human thermoregulatory system using
continuous-time block-oriented predictive modeling
Derrick Rollinsa, b,∗ , Nidhi Bhandarb , Sandra Hultinga
a Department of Statistics, Iowa State University, Snedecor Hall, Ames, IA 50011, USA
b Department of Chemical Engineering, Iowa State University, 2114 Sweeney, Ames, IA 50011, USA
Received 25 June 2004; received in revised form 20 July 2005; accepted 18 August 2005
Available online 2 November 2005
Abstract
This article presents preliminary results of a new research program for identifying predictive models for human thermoregulatory (HT)
response using only an individual’s attributes and their physical property data to build the model. This program is being developed in phases
and this article presents results of the first phase. This initial phase demonstrates that the proposed semi-empirical (i.e., gray-box), continuoustime, block-oriented modeling (BOM) approach [Rollins, et al. 2003. A continuous time nonlinear dynamic predictive modeling method for
Hammerstein processes. Industrial and Engineering Chemistry Research 42, 861–872; Bhandari and Rollins, 2003. A continuous-time MIMO
Wiener modeling method. Industrial and Engineering Chemistry Research 42, 5583–5595.] is capable of accurately predicting HT response.
This ability is demonstrated using real data from literature [Hardy and Stolwijck, 1966. Partitional calorimetric studies of man during exposures
to thermal transients. Journal of Applied Physiology 21(6), 1799–1806.] and computer generated data from a HT semi-theoretical model
with qualitatively accurate physiological behavior [Wissler, 1963. An analysis of factors affecting temperature levels in the nude human.
Temperature—its Measurement and Control in Science and Industry 3(3), 603–612; Wissler, 1964. Mathematical model of the human thermal
system. Bulletin of Mathematical Biophysics 26, 147–166.]. A critical strength of the proposed gray-box BOM approach is the use of physically
interpretable structures and model coefficients. This article discusses how this strength can be exploited to identify a predictive HT response
model for an individual without using environmental chamber data of the individual.
䉷 2005 Elsevier Ltd. All rights reserved.
Keywords: Thermoregulatory response; Predictive dynamic modeling; Semi-empirical modeling; Block-oriented systems; Process behavior
1. Introduction
The continued expansion of military, industrial, and scientific efforts in moderate and hostile environments suggests the
need for accurate prediction of human physiological response
under such conditions. This accomplishment can aid in the design of military chemical suits (Reneau et al., 1997), industrial
protective clothing (Reneau et al., 1999), and space suits. Furthermore, this accomplishment is also essential for the development of predictive control systems for environmental suits
as well as critically controlled environments such as space
∗ Corresponding author. Department of Chemical Engineering, Iowa State
University, Snedecor Hall, Ames, IA 50011, USA. Tel.: +1 515 294 5516;
fax: +1 515 294 2689.
E-mail address: drollins@iastate.edu (D. Rollins).
0009-2509/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ces.2005.08.036
ships and space stations. Thus, to advance physiological understanding, human thermoregulatory (HT) research has been
extensive, as described by Havenith et al. (1998). Phenomenological modeling in this context has meant explaining physiological behavior using scientific principles.
However, the scope of this article is system identification
and not phenomenological modeling. In this context, system
identification seeks to determine “how” input variables affect
HT response in contrast to phenomenological modeling that
also seeks to understand “why.” Moreover, our research program seeks to obtain HT predictive models for individuals
that depend on environmental input variables. These models
will be identified (i.e., developed) from existing knowledge,
and a subject’s personal attributes and physical property data.
Consequently, this information will be used in a unique way to
determine the three critical components of a predictive model:
the structures, explanatory variables, and the model parameters.
D. Rollins et al. / Chemical Engineering Science 61 (2006) 1516 – 1527
The model structures will come from library cataloged HT predictive models determined from other subjects. These models
will be cataloged in this library by physical attributes and environmental conditions. The model parameters will be determined
from individualized attributes and physical property data. For
an example of research in this area see Havenith (2001). The
critical explanatory variables will be obtained from knowledge
in the HT literature. For example, it is known from advances in
HT research that the following variables affect the human core
temperature: environmental temperature, wind speed, humidity, duration of exposure, etc., as well as individual attributes
such as age, gender, weight, height, surface area, level of fitness, and clothing. The concept of this HT model development
strategy for an individual is illustrated in Fig. 1.
Our fundamental modeling approach has been classified as
“semi-empirical” or “gray-box” modeling. Thus, we do not use
first principle modeling to obtain model structures (for examples see Gagge et al., 1986; Gordon et al., 1976). However,
unlike empirical models, since the models of this approach are
intelligent-based, their structures and the parameters can have
physical meaning and allow for extrapolation and independent
development of parameters. These properties of semi-empirical
modeling are exploited in this research program to build HT
models for individuals from limited experimentation. A critical
strength of our approach is its ability to define explicit structures for the static (i.e., ultimate), dynamic, and noise model
forms, that allows their independent identification and development. Thus, this attribute allows one to use knowledge and
information from several sources to build a model. This feature
is critical to our proposed HT system identification strategy for
individuals, as mentioned above. Another critical feature is the
continuous-time attribute that allows clear theoretical description of physical parameters such as time constants that can be
related to physical attributes of the person being modeled.
Our research program consists of several phases and this
article presents results of the first phase. In this initial phase,
1517
our goal is to demonstrate that the proposed, continuous-time,
block-oriented modeling (BOM) approach (Rollins et al.,
2003; Bhandari and Rollins, 2003) is capable of accurately
predicting HT response from measured inputs. We demonstrate this ability using data from two sources: real data from
literature (Hardy and Stolwijk, 1966) and computer generated
data from a widely accepted model with qualitatively accurate physiological behavior (Wissler model, 1963, 1964). The
Hardy and Stolwijck data are averaged values from three similar individuals. Our position is that, if our BOM can model
this behavior well, we see no reason why it should not be
able to model their individual behavior equally as well. Our
second study will treat the Wissler computer program as a
surrogate person. Given that this model adequately describes
HT behavior, at least qualitatively; this is adequate to meet the
data requirements of our study. Note that, a strength of this
study is the use of two vastly different types of data. Therefore, successful modeling in both cases will provide significant
evidence that our BOM approach is capable of capturing HT
behavior.
This article will specifically evaluate the feasibility of our
BOM method to model skin temperature and sweat rate in both
studies. In Section 2, we discuss BOM and present the proposed
approach. The real and simulated case studies are presented in
Sections 3 and 4, respectively. Finally, concluding remarks are
given in Section 5.
2. The proposed BOM technique
This section discusses general concepts of BOM and presents
the system identification methodology that we will use in this
work. Processes that can be described by series and/or parallel arrangement of nonlinear static blocks and linear dynamic
blocks are said to have block-oriented structures. One of the
simplest and most popular system is the Hammerstein structure, which consists of a nonlinear static gain block followed by
Environmental conditions
Wind
Temperature speed
Humidity
Blockstructure
Age
Gender
Subject
attributes
Weight
Exposure time
Library of models
Static gain and
dynamic
functions
Parameter
estimates
Semi-Empirical
HTRS model for
the subject
% Body fat
Auxiliary Information
(may require minor experimentation)
Fig. 1. The proposed library concept of this research program to identify BOMs for individual subjects using cataloged BOM structures, the environmental
conditions, and their attributes and physical property data.
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D. Rollins et al. / Chemical Engineering Science 61 (2006) 1516 – 1527
a linear dynamic block. Another popular system is the Wiener
structure which reverses this order. More specifically, each input first passes through a linear dynamic block followed by a
nonlinear static block. The Wiener structure allows each input
to have its own dynamic block while still addressing nonlinear
dynamics. For modeling Hammerstein systems, Rollins et al.
(2003) developed the Hammerstein Block-oriented Exact Solution Technique or H-BEST. Similarly, for modeling Wiener
systems, Bhandari and Rollins (2003) developed W-BEST.
BEST is a comprehensive system identification (i.e., modelbuilding) approach; its goals include: (a) maximum information from minimum trials, (b) identification of static, dynamic,
and error term model structures, and (c) accurate estimation
of all model parameters. Optimal statistical design of experiments (SDOE) is exploited to keep the number of experimental
runs (i.e., trials or step tests) to a minimum. Each experimental trial or design point is run from an initial steady state to a
final steady state to obtain rich dynamic and steady-state (ultimate response) information to allow separate determination of
static and dynamic model forms. Rollins and Bhandari (2004)
developed a constrained MIMO discrete-time method to exploit the advantages of discrete-time modeling. However, when
discrete-time methods are limited, the MIMO continuous-time
method developed by Bhandari and Rollins (2003) is also available. This method is available in “compact” form (using only
the most recent input changes) or classical form. The compact
form is restricted to step inputs (or piece-wise step inputs) and,
more critically, to reaching steady state between input changes.
Although there are no input restrictions for the classical form, it
is not compact and can depend on many previous input changes
to obtain the required accuracy. This article uses the classical
methods in both case studies since the number of input changes
is small. The classical prediction algorithm for H-BEST is presented below, followed by the one for W-BEST.
2.1. H-BEST model
For a Hammerstein system with input vector u(t) and single
output (t), we have
v(t) = f (u(t))
times t = 0, t1 , t2 , . . . , tK−1 .
⎧
u0 = 0 t = 0
⎪
⎪
⎪
u
0 < t t1
⎪
1
⎪
⎪
..
⎪ ..
⎨
.
.
u(t) =
,
u
tj −1 < t tj
⎪
j
⎪
⎪
⎪
..
..
⎪
⎪
.
⎪
⎩.
uK
tK−1 < t
(4)
where the input vector, uj = [u1,j , u2,j , . . . , up,j ]T such that
ui,j is the value of ith input in the j th interval. Based on
Eq. (4), V (s) becomes
f (u1 ) f (u2 ) − f (u1 ) −t1 s
+
e
+ ···
s
s
f (uK ) − f (uK−1 ) −tK−1 s
+
e
s
V (s) =
(5)
and
(t) = f (u1 )g(t)S(t) + (f (u2 ) − f (u1 ))g(t − t1 )
× S(t − t1 ) + · · · + (f (uK ) − f (uK−1 ))
× g(t − tK−1 )S(t − tK−1 )
K
=
(f (un ) − f (un−1 ))g(t − tn−1 )S(t − tn−1 ),
(6)
n=1
where f (u(t)) is the nonlinear static gain function and S(t) is
the unit step function. The dynamic function, g(t), is described
by Eq. (7) below:
1
g(t) = L−1 G(s)
.
(7)
s
2.2. W-BEST model
Similarly, for a Wiener system with multiple inputs and a
single output, we get
Vj (s)
= Gj (s),
Uj (s)
j = 1, 2, . . . , p
(8)
and
(t) = f (v(t))
(1)
⇒ vj (t) = L−1 {Gj (s)Uj (s)},
(9)
(10)
where v is the vector of intermediate hidden (i.e., unmeasured)
variable such that v = [v1 , v2 , . . . , vp ]T . Specifically for step
inputs, as described by Eq. (4):
and
(s)
= G(s)
V (s)
(2)
⇒ (t) = L−1 (G(s)V (s)),
(3)
where V (s) is the Laplace transform of v(t), L−1 is the inverse Laplace transform operator and G(s) is the linear dynamic transfer function of the process in the Laplace domain.
In both case studies, the inputs are step inputs, which can be
described using Eq. (4) below for K input changes occurring at
vj (t) = uj,1 gj (t)S(t) + (uj,2 − uj,1 )gj (t − t1 )S(t − t1 )
+ · · · + (uj,K − uj,K−1 )gj (t − tK−1 )S(t − tK−1 )
=
K
(uj,n − uj,n )gj (t − tn−1 )S(t − tn−1 ),
(11)
n=1
where f (·) is the nonlinear static function, and the dynamic
function, gj (t), is described by Eq. (12)
1
gj (t) = L−1 Gj (s)
,
(12)
s
D. Rollins et al. / Chemical Engineering Science 61 (2006) 1516 – 1527
where Gj (s) is the linear dynamic transfer function of the process, i.e., Gj (s) = Vj (s)/Uj (s). Thus, (t) is determined, as
in H-BEST, when f (·) and g(·) are found.
As we will show later, the error term in the model for the real
data study is serially correlated. Below, we present a procedure
to extend H- and W-BEST to address serially correlated noise
(i.e., the error term). This procedure first estimates the true
output, (t), under Model 1 (assuming “white” noise). Next,
it obtains the residuals and then their auto regressive, moving
average (ARMA) structure and initial estimates of ARMA parameters. It then pre-whitens the outputs under Model 2 using
the ARMA structure and re-estimates (t) (see Eq. (14) below).
More specifically, Model 1 is: y(t) = (t) + a(t), where at ∼
indep N(0, 2 ) for all t, and Model 2 is: y(t) = (t) + N (t),
where y(t) is the measured output value, and N (t) follows an
ARMA(p∗ , q ∗ ) structure (see Box and Jenkins, 1976) as shown
in Eq. (13):
Nt =
q ∗ (B)
at ,
p∗ (B)
(13)
∗
where q ∗ (B) = 1 − 1 B − 2 B 2 − · · · − q ∗ B q , p∗ (B) =
∗
1 − 1 B − 2 B 2 − · · · − p∗ B p , B is the backward difference
operator such that B r xt = xt−r , and ’s and ’s are the ARMA
parameters. In the pre-whitened form, under Model 2,
y(t) = (t) + 1 (y(t − t) − (t − t))
+ 2 (y(t − 2t) − (t − 2t)) + · · · + at
(14)
with estimator
ŷ(t) = ˆ (t) + ˆ 1 (y(t − t) − ˆ (t − t))
+ ˆ 2 (y(t − 2t) − ˆ (t − 2t)) + · · · ,
(15)
where
(B) =
p∗ (B)
q ∗ (B)
= 1 − 1 B − 2 B 2 − · · · .
(16)
We now give our system identification procedure formally by
the following six steps:
1. Select the SDOE and run the design (input changes) as a
sequence of step tests.
2. Average the steady-state data from each input change and
find the form of static gain function and estimate the parameters.
3. From a visual examination of the step tests, select the dynamic model forms (i.e., g’s) and estimate the dynamic parameters assuming Model 1 is true.
4. Then using the model residuals from Step 3, determine the
ARMA(p ∗ , q ∗ ) form of noise and initial estimates of the
p∗ + q ∗ parameters.
5. Simultaneously re-fit the dynamic parameters (using the
form and estimates found in Step 3 and the ARMA(p ∗ , q ∗ )
parameters (using the form and estimates found in Step 4)
under Model 2.
6. Check the residuals from Step 5 to verify white noise behavior.
1519
In the next section, we apply this procedure in the case study
using data from real subjects.
3. Case 1: BOM system identification in the real data study
There are a number of factors that need to be considered when modeling human thermoregulation. These can be
broadly grouped either as environmental factors (air temperature, relative humidity, wind velocity, etc.) or as individual
attributes (weight, height, age, gender, body fat, skin-type,
cardio-vascular condition, exercising level, etc.). For both case
studies, only environmental factors are considered as inputs due
to the restrictions imposed by the data we modeled. Likewise,
for the same reason, our studies are restricted to two outputs,
sweat rate and skin temperature. In this section, we present the
details of the steps involved in model building and the results
of model validation for the data obtained from experiments
by Hardy and Stolwijk (1966). The experiments were based
on the same protocol and used three male subjects of similar
build with the following average attributes: age—23.3 years,
height—1.83 m, weight—87.6 kg and surface area—2.02 m2 .
The purpose of the experiments was to measure the transient
response of sweat rate and skin temperature to sudden (i.e.,
step) changes in room temperature (T) and relative humidity
(H). Each experiment was 4 h long with a total of four experiments. In each experiment, the subjects were stationary in a
chamber with fixed temperature and humidity for 1 h. Then
they moved to a second chamber with different conditions and
stayed for 2 h. Finally, they went to a third chamber for the
last hour. The first three experiments were used for training the
model while the last experiment was used for testing or validating the model. The results reported for these experiments
were averaged responses for the three subjects. For more details about the experiment and data collection see Hardy and
Stolwijk (1966). Other researchers (Campbell et al., 1994;
Walker, 1999) have also used the sweat rate data from these
experiments. The results of these studies are compared with
the results of our model later in this article. We now present
the details of the BEST model-building procedure described
in the previous section.
Normally, the first step in our model-building approach is
the selection of input variables, the input or design space, and
the appropriate SDOE based on the á priori assumptions. Since
we are taking data from the literature, we go directly to Step
2 in the proposed system identification procedure described
in Section 2. The experimental input conditions and the input
design space for the training data are shown in Fig. 2 below.
As shown, the input changes are step tests as desired but the
input space is far from optimal in coverage.
For this study, both the input and the output variables were
converted to deviation variables. The second step involves the
estimation of the static function. The last three values of the
measured responses at the end of each step change were averaged and based on the input sequences in Fig. 2: there were
nine such step changes. The estimated nonlinear static functions were then obtained using linear regression as shown in
1520
D. Rollins et al. / Chemical Engineering Science 61 (2006) 1516 – 1527
Rel. Humidity
Room Temperature
45
45
43
41
39
37
35
33
31
29
27
25
25
Relative Humidity (%)
Temperature (deg C),
Rel. Humidity (%)
50
40
35
30
25
0
100
200
300
400
500
600
700
Train
Test
30
Time (min)
35
40
45
50
Room Temperature (deg. C)
0.6
0.6
0.5
0.5
Partial Auto Correlation Function
Auto Correlation Function
Fig. 2. The sequence of step changes in the room temperature and relative humidity for training cases (left plot) along with the input space for training and
testing (right plot). The data are from Hardy and Stolwijk (1966).
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Lag
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Lag
Fig. 3. The ACF and PACF for residuals from Model 1 for sweat rate response. These results indicate AR(1) behavior.
Eqs. (17) and (18) below:
fˆsweat (T , H ) = 0.021 + 0.010T − 0.008H
+ 0.0006H 2
models the dynamic effects of room temperature; the second
one models dynamic effects of relative humidity. These fitted
dynamic models for the sweat rate response are both first-order
and are given in Eqs. (19) and (20) below:
(17)
fˆTskin (T , H ) = 0.239T + 0.174H − 0.0074H 2 , (18)
where T is the deviation variable for room temperature, H
is the deviation variable for relative humidity, and fˆSweat (·) and
fˆTskin (·) are the estimated static functions for sweat rate and
skin temperature, respectively, in mg/cm2 / min and ◦ C. Note
that dividing Eq. (17) by 6000 converts it to SI units kg/m2 /s.
The next step is the identification of the dynamic models
by assuming a certain block-oriented structure. For this case,
we tried both Wiener and Hammerstein structures and found
that the Wiener structure produced a more accurate fit of the
sweat rate response while the Hammerstein structure produced
a more accurate fit for the skin temperature response. Since
the sweat rate response relies on two inputs and uses a Wiener
structure, two dynamic models must be estimated. The first one
ĝ1,sweat (t) = 1 − e
ĝ2,sweat (t) = 1 − e
− ˆ t
s,1
− ˆ t
s,2
,
(19)
,
(20)
where ˆ s,1 and ˆ s,2 are the estimated time constants and their
estimates are 13.17 and 15.13 min, respectively. For the skin
temperature response, we used a Hammerstein structure, and
thus, only one dynamic function is required for modeling. This
fitted dynamic function is of second-order, critically damped,
form and is given in Eq. (21):
(t)
t
−
ĝTskin (t) = 1 − 1 +
e ˆ T ,
ˆ T
(21)
where ˆ T is the estimated time constant with ˆ T = 4.05 min.
D. Rollins et al. / Chemical Engineering Science 61 (2006) 1516 – 1527
0.25
0.04
0.2
0.03
0.15
Residuals from Model 2 .
Auto Correlation Function
1521
0.1
0.05
0
-0.05
-0.1
-0.15
0.02
0.01
0.00
-0.01
-0.02
-0.03
-0.2
-0.25
-0.04
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Lag
0
100
200
300
400
Time (min)
500
600
700
Fig. 4. The ACF and time series plot of the residuals from Model 2 showing white noise behavior.
Sweat rate (Kg/m2/s) 10^6
60.00
Data
Eta-hat
Y-hat
50.00
40.00
30.00
20.00
10.00
0.00
0
100
200
300
400
Time (min)
500
600
700
300
400
Time (min)
500
600
700
37.5
37.0
Skin Temperature (deg C)
In the fourth step, using the residuals from Model 1, we:
(1) obtained the auto correlation function (ACF) and partial auto correlation function (PACF); (2) determined the
ARMA(p ∗ , q ∗ ) structure of the noise; and (3) calculated the
initial parameter estimates. This was done using the Minitab
statistical software package. For space consideration, only the
ACF and PACF for the residuals for sweat rate response are
shown (Fig. 3). Based on the ACF and PACF, and using the
ARIMA command in Minitab, the residuals for sweat rate were
modeled as an auto regressive, order one (i.e., AR(1)) process
(see Box and Jenkins, 1976) with an initial parameter estimate,
ˆ = 0.54. More specifically, ˆ 1 in Eq. (15) is ˆ = 0.54, and
ˆ i = 0 for i not equal to 1. The residuals for skin temperature were modeled as an AR(2) process with initial values of
ˆ 1 =0.50 and ˆ 2 =0.22. More specifically, ˆ 1 = ˆ 1 =0.50, and
ˆ 2 = ˆ 2 = 0.22, and ˆ i = 0 for i not equal to 1 or 2 in Eq. (15).
In the fifth step, using the dynamic models and noise model
obtained in previous steps, we re-estimated the dynamic parameters as well as the AR parameters. The new estimated dynamic parameters are: ˆ 1,s = 13.67 and ˆ 2,s = 13.52 for the
sweat rate response and ˆ T = 4.25 for skin temperature. Also
for the sweat rate response, the revised estimate for the AR parameter is ˆ = 0.55, while the revised estimates for the skin
temperature response are ˆ 1 = 0.52 and ˆ 2 = 0.26.
The final step is to make sure that the residuals from Model
2 are white noise or uncorrelated. This can be seen from Fig. 4,
where none of the lags are highly significant and the residuals
are uniformly scattered about zero.
The fitted models for the training data are shown in Fig. 5
below for the sweat rate and the skin temperature responses.
The estimator that depends only on the inputs is given by Etahat (ˆ) and the one-step-ahead (OSA) predictor is given by
Y-hat (ŷ). The proposed method fits both the responses quite
well in training as seen in this figure.
The fitted models were then evaluated using the test
sequence shown in Fig. 6. As mentioned previously, other researchers have modeled the data for sweat rate as well; comparisons of these models along with our proposed model against
true response data are shown in Fig. 7. These previous models
Data
Eta-hat
Y-hat
36.5
36.0
35.5
35.0
34.5
34.0
33.5
33.0
32.5
0
100
200
Fig. 5. The fitted sweat rate (R 2 = 1.00 for Y-hat and Eta-hat) and skin
temperature (R 2 =0.97 for Y-hat and 0.96 for Eta-hat) response to the training
sequence given in Fig. 2.
include the ANN model proposed by Campbell et al. (1994)
and the two empirical models, ANN and PLS (projection to
latent structures), developed by Walker (1999) in addition to
the OSA (i.e., Y-hat) predictor for the proposed approach.
Fig. 7 compares the four fits on the basis of the statistic rfit ,
which is the correlation coefficient between the observed values of the output and fitted response. This is a common way of
Temperature (deg C), Humidity (%)
1522
D. Rollins et al. / Chemical Engineering Science 61 (2006) 1516 – 1527
45
Room Temperature
Table 1
SSPE results for the methods in Fig. 7
Rel. Humidity
Method
SSPE
Relative SSPE
Proposed (Y-hat) (OSAP)
Proposed Eta-hat
PLS-Walker
ANN-Campbell et al.
ANN-Walker
0.024
0.038
0.091
0.084
0.053
1.00
1.63
3.85
3.56
2.23
40
35
30
25
0
40
80
120
Time (min)
160
200
is the SSPE for the OSAP of the proposed method. From this
table, one sees that the relative SSPE’s are from about 2–4 times
greater than the SSPE for the OSAP of the proposed method
and that the SSPE for the simulation prediction (Eta-hat) of the
proposed method is much less than the other methods.
The skin temperature response for the test data, along with
the sweat rate response of the proposed method is shown in
Fig. 8. In both cases, rfit for the OSAP is 0.97 and it is 0.96
for Eta-hat for sweat rate. We did not include Eta-hat for skin
temperature because we did not obtain comparative results
with other methods. Thus, based on the results presented in
Figs. 7 and 8, and Table 1, we conclude that the proposed
method captured the physiological thermoregulatory behavior
of this data for skin temperature and sweat rate quite well. In
240
Fig. 6. The test sequence for evaluation of fitted models in Figs. 7 and 8.
The data are from Hardy and Stolwijk (1966).
assessing fit as discussed by Devore (2004). As shown, the proposed method has the highest rfit value of 0.97 indicating that it
fit the data the best. Another assessment of the results in Fig. 7
is given in Table 1 that compares the sum of squared prediction
error (SSPE) and the relative SSPE for the four fits. The prediction error for each observation is defined as the observed value
minus the predicted value. The relative SSPE is determined by
dividing each SSPE by the smallest SSPE, which is this case,
60.00
Measured Data
Proposed (Y-hat)
50.00
Sweat rate (Kg/m2/s)x10^6
Sweat rate (Kg/m2/s)x10^6
60.00
40.00
30.00
20.00
10.00
0.00
Measured data
PLS - Walker
50.00
40.00
30.00
20.00
10.00
0.00
0
40
80
120
160
200
240
0
40
80
Time (min)
160
200
240
Time (min)
60.00
60.00
Measured data
ANN-Campbell et al.
50.00
Sweat rate (Kg/m2/s)x10^6
Sweat rate (Kg/m2/s)x10^6
120
40.00
30.00
20.00
10.00
0.00
Measured data
ANN-Walker
50.00
40.00
30.00
20.00
10.00
0.00
0
40
80
120
Time (min)
160
200
240
0
40
80
120
160
200
240
Time (min)
Fig. 7. The predicted sweat rate responses for the proposed method (rfit = 0.97), PLS-Walker (rfit = 0.92), ANN-Campbell et al. (rfit = 0.93) and ANN-Walker
(rfit = 0.94) using the input test sequence shown in Fig. 6. The proposed method is shown in the upper left corner. The data are from Hardy and Stolwijk (1966).
D. Rollins et al. / Chemical Engineering Science 61 (2006) 1516 – 1527
Table 2
The coded values for the inputs
Sweat rate (Kg/m2/s) 10^6
60.00
Data
Eta-hat
Y-hat
50.00
Input (units)
40.00
30.00
Temperature of environment (T) (◦ C)
20.00
Wind speed (W) (mps)
Humidity (H) (%)
Coded level
−1
0
1
32.2
0.45
75
34.4
1.34
80
36.7
2.24
85
10.00
0.00
0
50
100
150
200
Time (min)
37.0
Data
Eta-hat
Y-hat
36.5
Skin Temperature (deg C)
1523
36.0
35.5
35.0
34.5
34.0
33.5
33.0
32.5
32.0
0
40
80
120
160
200
240
Time (min)
Fig. 8. The predicted sweat rate (rfit = 0.97 for Y-hat and rfit = 0.96 for
Eta-hat) and skin temperature (rfit =0.97 for Y-hat) responses for the proposed
method (BEST) using the test sequences in Fig. 6. The data are from Hardy
and Stolwijk (1966).
the next section we present the results of the Wissler data study,
which uses an SDOE for the training input changes and adds
another input, wind speed. Thus, this study fully applies our
system identification method described in Section 2.
4. Case 2: BOM system identification in the Wissler data
study
In this section, we present the results of the model-building
procedure using data from the Wissler computer program. All
the details of this program have not been included for space
consideration (see Wissler, 1963, 1964 for details). The set of
initial conditions for the Wissler code pertain to a man in a
sitting position wearing a cotton shirt and pants, with a weight
of 82.6 kg, a skinfold thickness 0.012 m, and a resting metabolic
rate of 83.7 J/s, in an environment with a temperature of 26.7 ◦ C,
an air pressure of 1.01 × 105 Pa (1 atm), a relative humidity of
50%, and a wind speed of 1.34 m/s.
As mentioned in Section 2, the first step of the modeling
procedure is the selection of the experimental design including
the input variables. However, for simplicity, the input variables
chosen in this study were restricted to the temperature of the
environment (T), the relative humidity of the environment (H),
and the wind speed of environmental air (W). The upper and
lower limits of the input variables in this study were chosen
from a realistic perspective so that they would cover as wide
a range as possible without adversely affecting the subject.
This is critical for experiments on human subjects. Therefore,
to be able to observe changes in the sweat-rate response, the
temperature lower limits of the input variables in this study
were chosen from a realistic perspective so that they would
cover as wide a range as possible without adversely affecting
the subject. This is critical for experiments on human subjects.
Hence, to be able to observe changes in the sweat-rate response,
the temperature of the environment should be higher than the
body temperature. In these situations, the loss of body heat can
become very dependent on sweat evaporation, which in turn
depends on the relative humidity of the surrounding air.
When the ambient humidity is high, the capacity of the environment to accept water is reduced and so the sweat evaporation rate is also reduced. Hence, for this study, the experimental
region consisted of high temperatures, high relative humidity,
and low wind speeds; i.e., conditions where the responses are
highly affected by input changes.
The selection of an appropriate SDOE to optimize the information content of the data for estimating model parameters depends on the á priori assumptions about the significant
main effects and interaction terms of the inputs in the nonlinear
static gain functions, i.e., the f (·)’s. In this study, we assume
that only the second-order effects (i.e., the quadratic terms and
two-factor interactions) are significant in the static gain (or the
ultimate response) functions for the outputs.
There are a variety of designs that allow estimation of secondorder effects, such as three-level factorial designs, central composite designs (CCD) and Box Behnken designs (BBD), to
name a few. The goal is to select a design with few runs and still
obtain accurate estimates for the main effects and interaction
terms in the model. A complete factorial design that models all
possible interactions with three inputs at three levels requires
33 = 27 trials. A corresponding CCD would require a total of
15 experimental trials, whereas a BBD needs 13 trials without replicating the center point. Since we are using computergenerated data for this study, there is no need for replicated
runs because there is no noise in these values.
Since the BBD has the least number of trials or runs, it
was chosen for this study. The three input levels are coded as
−1, 0, 1 for each input variable, and the actual values associated with these levels are shown in Table 2. The environmental
temperature ranges from 32.2 to 36.7 ◦ C, the wind speed from
0.45 to 2.24 meters per second (mps), and the relative humidity from 75% to 85%. The 13 experimental trials for the BBD
are shown in Table 3. These trials were generated using the
1524
D. Rollins et al. / Chemical Engineering Science 61 (2006) 1516 – 1527
Table 3
The design points based on BBD
Run #
T (C)
W (mps)
H (%)
1
2
3
4
5
6
7
8
9
10
11
12
13
34.4
34.4
36.7
34.4
36.7
32.2
32.2
36.7
32.2
34.4
36.7
34.4
32.2
0.45
1.34
1.34
0.45
1.34
1.34
0.45
2.24
2.24
2.24
0.45
2.24
1.34
85
80
5
75
75
75
80
80
80
75
80
85
85
software package JMP, version 4.0.4, from SAS Institute Inc.,
on a PC platform. For this study, data were obtained by executing the 13 experimental trials provided by the BBD by running the Wissler code with the temperature, humidity, and wind
speed set to the values listed in Table 3 and letting the responses
approach steady state for each run. This time was about 80 min
for the skin temperature but about 360 min for the sweat rate
with sampling every 10 min.
The second step is to use the steady-state data to find the
nonlinear static gain function for each of the responses using multiple regressions. From the data collected, we have 13
steady state values for both outputs (responses). The static gain
functions identified for the skin temperature and sweat rate are
shown in Eqs. (22) and (23), respectively, with all terms significant at the 0.05 level. The proportion of variability (i.e., R 2 )
in the data explained by these static gain models is 99% and
98% for skin temperature and sweat rate, respectively.
fˆTskin (T , H, W ; ˆ T )
= 0.81 + 0.29T + 0.013H − 0.15W − 0.0056T 2
− 0.0011T H + 0.0094T W ,
(22)
fˆSweat (T , H, W ; ˆ S )
= 4.83 − 0.54T − 0.13H − 0.091W
+ 0.017T 2 + 0.011T H ,
where T , H , and W are deviation values for temperature,
relative humidity, and wind speed from their initial conditions
of 26.7 ◦ C, 50%, and 1.34 mps, respectively, fˆTskin (·) gives the
change in skin temperature in ◦ C, fˆSweat (·) gives the change in
sweat rate in g/min (to convert to kg/s divide by 60000), and
ˆ T and ˆ S are the estimated parameter vectors. We see from
Eqs. (22) and (23) that the quadratic effects associated with the
temperature is significant for both the responses as well as the
interaction effects of humidity and temperature. In addition, for
the skin temperature, the interaction effect of temperature and
wind speed is also significant.
The third step in model building is the identification of the
dynamic functions. This identification involves trial and error
and visual inspection of the transient responses can aid in the
selection of these model forms. The output responses are shown
in Fig. 9 for both the skin temperature and the sweat rate responses for a single run (i.e., Run 3). Based on these plots,
an over-damped, second-order-plus-lead model form was selected for the skin temperature response and a critically damped,
second-order-plus-lead with dead-time model was selected for
the sweat rate. The model forms for these dynamic transfer
functions are given below in Eqs. (24) and (25), respectively.
(ˆT a − ˆ T 1 ) −t/ˆT 1
e
(ˆT 1 − ˆ T 2 )
(ˆT a − ˆ T 2 ) −t/ˆT 2
+
e
,
(24)
(ˆT 2 − ˆ T 1 )
ˆ
)
ˆ Sa − ˆ S )
− (t−
ˆ
ˆ
ĝSweat (t; ˆ S )= 1 +
(t − ) − 1 e S ,
ˆ 2S
(25)
ĝTskin (t; ˆ T )=1+
where ˆ T = [ˆT 1 , ˆ T 2 , ˆ T a ]T is the estimated parameter vector for the dynamic model for skin temperature, and ˆ S =
ˆ T is the estimated dynamic parameter vector for the
[ˆS , ˆ Sa , ]
36.5
60
Sweat rate (Kg/s)x10^6
36
Skin Temperature (°C)
(23)
35.5
35
Wissler data
Eta-1-hat
34.5
34
33.5
33
32.5
50
40
Wissler data
Eta-1-hat
30
20
10
0
32
0
50
100
150
Time (min)
200
250
0
50
100
150
200
250
300
Time (min)
Fig. 9. The fitted skin temperature and sweat rate response for one of the runs (Run 3) of the BBD for the Wissler simulation data.
350
85
45
84
40
Humidity (%)
83
35
82
30
81
80
H (%)
T (C)
W (mps)x10
79
78
25
20
15
77
10
76
75
Wind speed (mps), Temp (C)
D. Rollins et al. / Chemical Engineering Science 61 (2006) 1516 – 1527
5
0
50
100
150
Time (min)
200
Fig. 10. The test sequence for the study using Wissler data.
sweat rate model. The mean estimated dynamic parameters we
obtained for the skin temperature are: ˆ T 1 = 1.75 min, ˆ T 2 =
40.69 min, and ˆ T a = 14.27 min. The estimates for other dynamic parameters for the sweat rate are ˆ S = 50.94 min and
ˆ Sa = 32.81 min. The sweat rate response exhibited a delay and
therefore needed dead time in the dynamic model, as shown
in Eq. (26). While the estimates of the dynamic parameters,
ˆ S and ˆ Sa , do not change significantly between the runs, the
estimate for dead time does change from run to run. This can
be attributed to the fact that for runs with a low heat index, a
subject will take more time to start sweating than for runs with
a higher heat index. The semi-empirical approach used here
allows the modeler to account for this variation using the estimates for dead time from the 13 runs and by fitting a multiple
regression model as shown in Eq. (26). The terms in this model
are all significant at the 0.05 level.
ˆ = 376.06 − 33.83T + 16.31W + 0.82T 2
− 0.94T W .
(26)
Fig. 9 show the observed skin temperature and sweat rate
data for Run 3 along with the fitted dynamic models given by
Eqs. (24) and (25). These fits are quite good. Note that, in
practice, these model forms would be developed from data collected on a subject placed in an environmental chamber.
For this study, we limited the model building steps to
obtaining the estimators based only on the inputs (Eta-hat)
since we did not add noise. The validation of the developed
BEST models is performed by running another sequence of
arbitrary input changes (shown in Fig. 10) and comparing
the predicted responses with those observed from the Wissler
model.
The Wissler response data and the predictions by the proposed method for skin temperature (rfit = 0.98) and sweat rate
(rfit = 0.99) are shown in Fig. 11. As shown, the predictions
from the model agree well with the Wissler response data. Thus,
the proposed method appears to capture the HT behavior expressed by the Wissler program quite well.
5. Concluding remarks
This article presented the concept of our overall program
in human thermoregulatory (HT) system identification for obtaining predictive models for individual subjects. This program
use block-oriented model (BOM) forms and this work demonstrated that our BOM approach is able to capture HT behavior
as demonstrated in two modeling cases. One case used experimental data from Hardy and Stolwijk (1966) and the other case
used data from the Wissler (1963, 1964) computer program
as a surrogate human. The models in the proposed approach
have phenomenological characteristics and the parameters have
physical meaning, which can be related back to the attributes of
the subject. It is our ultimate goal to exploit these characteristics in HT system identification to obtain individualized models
requiring only the attributes and physical property data of the
subject. This article represents the first phase of our research
to reach this objective.
Before we can realize our ultimate goal, we will need to
create a library of model structures over a broad input space
as discussed in Section 1 of this article. Therefore, a major
effort of future research will consist of developing this library.
In terms of efficient modeling, there are two issues that we
will need to address: (1) the number of experimental trials;
40
Skin Temperature (°C)
Sweat rate (Kg/s)x10^6
Wissler data
30
Eta-1-hat
20
10
0
0
50
100
150
Time (min)
200
1525
36.5
36
35.5
35
34.5
34
33.5
33
32.5
32
31.5
Wissler data
Eta-1-hat
0
50
100
150
Time (min)
200
Fig. 11. The predicted and observed (i.e., Wissler) responses for sweat rate (rfit = 0.99) and skin temperature (rfit = 0.98) to the input test sequence in Fig. 10.
1526
D. Rollins et al. / Chemical Engineering Science 61 (2006) 1516 – 1527
and (2) the length of time for each experiment. Our modeling approach will be useful in helping us to meet these
challenges. More specifically, since our approach is embedded in optimal statistical design methodology, we will exploit this feature to minimize the number of runs and the
time for each run when developing models from test subjects. We plan to present results of this research in the near
future.
Notation
AM
B
CpM
f
g
H
K
MM
rfit
R2
s
t
T
u
U
W
y
surface area of metal block
backward difference operator
heat capacity of the metal block
static gain function
linear dynamic function
relative humidity of the environment
process gain parameter
mass of the metal block
test data correlation coefficient for the observed values and the fitted values
proportion of fitted variability explained by the
model
seconds
time
temperature of the environment
vector of input variables
overall heat transfer coefficient
wind speed of ambient air
output (i.e., observed) variable
Greek letters
i
t
a
parameter i in the empirical model
vector of parameters in the nonlinear static gain
functions
sampling time
expected value of y or true output value
dead time for the dynamic function of the
sweat rate response
parameter in noise model
time constant
dynamic lead parameter
vector of parameters in the dynamic functions
ARMA parameter
Subscripts
A
E
p
p∗
q
q∗
ref
Sweat
air
environment
number of inputs
number of AR parameters
number of outputs
number of MA parameters
reference
sweat rate
Tskin
W
skin temperature
water
Superscript
∧
estimate
Abbreviations
ACF
ANN
AR(p)
ARMA
BBD
BEST
BOM
CCD
H-BEST
HT
MIMO
OSA
PACF
PLS
SDOE
SISO
SSE
W-BEST
auto correlation function
artificial neural networks
auto regressive of order p
auto regressive moving average
Box-Behnken design
block-oriented exact solution technique
block-oriented modeling
central composite design
Hammerstein block-oriented exact solution
technique
human thermoregulatory
multiple input multiple output
one-step-ahead
partial auto correlation function
projection to latent structures
statistical design of experiments
single input single output
sum of squared errors
Wiener block-oriented exact solution technique
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