24
Case Studies
Selected case studies on time-series modeling, identification of simulation and real-life pro-
cesses are presented. The objective is to demonstrate the principles and practical aspects of
identification outlined in this text.
24.1 ARIMA MODEL OF INDUSTRIAL DRYER TEMPERATURE
24.1.1 PROCESS DATA
The case study is concerned with developing a pure time-series model for the outlet temperature
of a dryer in an industrial process. A cascade controller regulates this temperature at a set-point of
T = 134
C with the help of PI controllers in both master and slave loops.
The time-series model that we develop here can be used for predicting the eects of disturbance
and in assessing the performance of the temperature control loop.
24.1.2 BUILDING THE TIME-SERIES MODEL
In developing a time-series model for the series, we follow the general procedure outlined in Table
19.2.
Step 1: Figure 24.1(a) shows the time series obtained at T
s
= 15 second intervals. The auto-
correlation function in Figure 24.1(b) shows a slow decay suggesting the presence of a long-
memory characteristic at the scale of observation. Further, the PACF in Figure 24.1(c) has a
near-unity value at lag l = 1 indicating the possibility of a unit root in the AR component.
The spectral density estimated by Welch’s smoothed periodogram is shown in Figure 24.1(d)
shows significant power at ω = 0 and neighboring frequencies, corroborating the presence of
integrating eects. Additionally, a peak in the p.s.d. at ω = 0.055 rad/sample (normalized fre-
quency) is a clear indication of a periodicity in the series. From a performance assessment view-
point this is an indication of poor performance.
Note that although the actual process may not strictly contain an integrating eect, the statistics
clearly indicate the suitability of a model with a pole on the unit circle. To verify this claim, a
unit root test can be conducted on the series. Leaving this exercise to the reader, we adopt a less
rigorous route. For the purpose of modeling, we select the first N = 2000 observations and mean
center it. There is no visible polynomial trend in the series (a unit root test can be conducted to
confirm this hypothesis).
Fitting an AR(1) model to the training data results in the noise model
ˆ
H
1
(q
1
) =
1
1 0.9966
(±0.0019)
q
1
(24.1)
where the number in the parentheses is the standard error in the estimate. Clearly the data presents
a case for dierencing.
Step 2: Does not apply to the case study.
Step 3: Does not apply to the case study.
718
Case Studies 719
0 500 1000 1500 2000 2500 3000
128
130
132
134
136
138
140
142
144
146
Time
Temperature
(a) Temperature series
0 20 40 60 80 100
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
ACF
Lags
(b) Auto-correlation function
0 5 10 15 20
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
PACF
Lags
(c) Partial auto-correlation function
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0
5
10
15
20
25
30
35
40
Normalized frequency (rad/sample)
Spectral density
(d) Power spectral density
FIGURE 24.1 Series, ACF, PACF and spectral density of the temperature series.
Step 4: A first-degree of dierencing d = 1 is deemed appropriate. The dierenced series and its
ACF are shown in Figures 24.2 and 24.1(b), respectively.
There is no visible evidence of non-stationarity (again one could conduct a unit root test here).
Further, it is insightful to note that the variance of the dierenced series is σ
2
v
= 0.0336 whereas
the variance of the original series is σ
2
v
= 4.3794. Thus, a single dierencing has explained about
99% of the variance. The predictability of the dierenced series remains to be analyzed.
The power spectral density post-dierencing shows zero to very low power at ω = 0 indicat-
ing absence of any integrating eects. With the earlier dominant integrating eect taken away,
additional peaks in the p.s.d. are now visible pointing to more than one periodic component. At
this juncture one could first model the periodic components separately followed by a time-series
modeling of the residual. Alternatively one could directly model the dierenced series expecting
the time-series model to capture both periodic and stochastic eects. We shall adopt this route
for this case study. The final model should have imaginary components in order to explain the
oscillatory characteristics.
Step 5: With the above arguments we begin with an ARMA(2,1) model. The NLS estimate of the
model is obtained as
ˆ
H
21
(q
1
) =
1
(±0.0669)
0.5085 q
1
1 1.017
(±0.0725)
q
1
+ 0.111
(±0.057)
q
2
,
ˆ
σ
2
e
= 0.0174 (24.2)
However, this model is unsatisfactory for several reasons: (i) the parameter estimate
ˆ
d
2
is in-
significant, (ii) the poles of the identified model are purely real and (iii) residuals have significant
correlation in them as confirmed by Figures 24.3(a) and 24.3(b) showing the residual-ACF and
the Box-Ljung Q statistics, respectively.

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