A new method to estimate a first-order plus
time delay model from step response
Giuseppe Fedele ∗
Dipartimento di Elettronica Informatica e Sistemistica,
Università degli Studi della Calabria,
87036, Rende (Cs), Italy
Abstract
In this paper an identification method to estimate the parameters of a first order plus
time delay model is proposed. Such a method directly obtains these parameters using
a new linear regression equation. No iterations in calculation are needed. A simple
true/false criterion to establish if the hypothesis on the process type is correct can be
easily derived. The proposed method shows an acceptable robustness to disturbance
and measurement noise as it is confirmed by several simulated experiments.
Key words: First order plus time delay model, Algebraic identification,
Least-squares method
1
Introduction
A large number of industrial plants is modeled by a first order plus time delay
(FOPTD) transfer function as follows:
G(s) =
K
e−Ls .
Ts + 1
(1)
The wide use of model (1), especially in PID controller, is due both to its simplicity and its ability to capture the essential dynamics of several industrial
processes [2]. Several identification procedure have been proposed to identify
the parameters K, T and L in model (1). Although the recent literature insists
∗ Tel.: +39-0984-494720; fax: +39-0984-494713.
E-mail address: fedele@si.deis.unical.it (G. Fedele)
Preprint submitted to Journal of The Franklin Institute
on closed-loop identification schemes, see for example [5], [7] and [6], the methods based on the measurement of the process step response, see [3],[8] and the
references within, are still important because of the simple physical interpretation, and their easy implementation in industrial environments. The most
used methods are: graphical methods which determine model parameters directly from step response curves, and area-based methods. Graphical-methods
are quite sensitive to measurement noise. In these methods the process static
gain K is obtained from the steady state of the process input and output. The
intercept of the tangent to the step response that has the largest slope with
respect to the horizontal axis gives L. T is determined from the difference
between L and the time when the step response reaches the value 0.63K. In
the area-based methods the process static gain K is estimated firstly from
the steady states of the process input and output. Then the average residence
time Tar is computed as
R∞
(y(∞) − y(t))dt
Tar =
0
= L + T,
K
where y(t) is the step response of the process. Finally T and L are estimated
as follows:
e
T =
TRar
y(t)dt
0
K
L = Tar − T.
,
In view of practical applications, the main drawback of the area-methods
is that the identification precision is highly dependent on the calculation of
areas, which means that the step must be long enough for the process to
enter the new steady state completely. A recent approach in [3,8,9] provides
a least-squares method to estimate the parameters of the continuous FOPTD
model. Unfortunately the main drawback of such approach is that the first
sample of the process output y(t1 ) should not be taken into the algorithm until
t1 ≥ L, when the output deviates from the previous steady state. Therefore the
method needs the process output to be monitored for a period, the “listening
period”, during which the noise band Bn can be found. Then the step test
starts at time τ where the output y satisfies
y(τ ) > 2Bn .
The method here proposed is based on new linear regression equations with redundant variables, and uses filtered signals obtained directly from the process
output. The approach uses the algebraic derivative method in the frequency
domain [1,4], yielding exact formulas, when placed in the time domain, for
the unknown parameters: dead time and constant time. Due to use of the
2
process output integrals in the regression equations, the resulting parameter
estimation is very robust in the face of large measurement noise in the output.
Implicit in the method is a criterion to establish the correctness of the hypothesis that the process is adequately represented by the FOPTD model. Another
feature is that a modified area-based method can be derived directly from our
equations. Simulations in noisy environment show the effectiveness of the proposed method. The paper is organized as follows. In section 2, the proposed
method and an optimization procedure to solve the regression equations are
presented. Section 3 gives simulation results on some typical processes. The
last section is devoted to conclusions.
2
The proposed method
Suppose the given process is under zero initial condition, before a step change
at t = 0 in the process input u(t) is applied. If the step amplitude is U0 then
the output of process (1) satisfies the equation
(T s2 + s)Y (s) = KU0 e−Ls .
(2)
Eq. (2) is differentiated one time with respect to the complex frequency s, and
the result is divided by L to obtain:
"
#
dY (s)
1
(2T s + 1)Y (s) + (T s2 + s)
= −KU0 e−Ls .
L
ds
(3)
To eliminate the term KU0 e−Ls eq. (2) and (3) are combined together:
(T s2 + s)LY (s) + (2T s + 1)Y (s) + (T s2 + s)
dY (s)
= 0.
ds
(4)
The division by s2 allows to eliminate all the derivations implicit in the multiplication by power of s:
h
T 2s−1 Y (s) +
dY (s)
ds
i
+ Ls−1 Y (s) + LT Y (s) = −s−2 Y (s) − s−1 dYds(s) .
(5)
When the inverse Laplace transform is applied to eq. (5), one has
f1 (t)T + f2 (t)L + f3 (t)θ = f4 (t)
3
(6)
where
Rt
f1 (t) = 2 y(σ)dσ − ty(t),
Rt
0
f2 (t) = y(σ)dσ,
(7)
0
f3 (t) = y(t),
f4 (t) = −
and θ = LT .
Rt Rσ
Rt
y(λ)dλdσ + σy(σ)dσ.
0
0 0
Note that fi (t), i = 1, ..., 4 can be regarded as the outputs of a system whose
Simulink diagram is shown in Fig. 1.
Fig. 1. Simulink diagram for fi (t), i = 1, ..., 4.
The linear equations (6) and (7) allow us to determine the unknown process
parameters T and L and the value of an auxiliary variable θ. Assume that
fi (t), i = 1, ..., 4 are measured at times 0, Ts , ..., (n − 1)Ts , where Ts is the
sampling period and n is the number of samples, and let define
x = [T L θ]T ,

and



Ψ=



f1 (0)
f2 (0)
f3 (0)
f1 (Ts )
..
.
f2 (Ts )
..
.
f3 (Ts )
..
.
f1 ((n − 1)Ts ) f2 ((n − 1)Ts ) f3 ((n − 1)Ts )

then eq. (6) implies
(8)




γ=




f4 (0)
f4 (Ts )
..
.
f4 ((n − 1)Ts )
Ψx = γ.
4




,



(9)





,




(10)
(11)
When real process data are used to build Ψ and γ with n 3, eq. (11) has
no solution, and an estimate x̂ of the parameters is obtained by least-squares
as
−1
ΨT γ.
(12)
x̂ = ΨT Ψ
If the actual measurement of the process output is corrupted by measurement
noise, which is assumed to be a white noise, it follows from [10] that in such
case, the ordinary least-squares estimate in (12) is not consistent, because the
term due to the noise is now a correlated noise instead of a white noise. One
solution is to use the instrumental variable least-squares method [11].
Usually in process control, the unknown parameter K is directly estimated
from the steady-state output as K = y(∞)/U0 . However, there do exist such
cases that the step stops before the steady state is reached. Then, one has to
estimate the steady-state K from the step test as well, instead of reading it
from the steady output. In order to estimate the unknown parameter K the
process output transient y(t) after t = L is considered:
y(t) = KU0 1 − e−
t−L
T
,
t ≥ L.
(13)
Integrating y(t) from t = 0 to t = τ > L gives
Z
0
τ
y(t)dt = U0 K τ − L − T + T e−
τ −L
T
.
(14)
The parameter K can be estimated by eq. (14), with y(t) given by the process
output, as
K̂ =
Rτ
y(t)dt
0
U0 τ − L − T + T e−
τ −L
T
,
τ > L,
(15)
where for L and T we consider the estimated parameters.
It must be stressed that no other test or further computations are required
to determine the unknown parameter K, since the integral term in eq. (15) is
directly given by f2 (τ ) which is already computed. Although one also has to
use some data after t = L, in estimating K, there is no additional experimental
work and the computation is rather efficient.
Remark 1 Eqs. (5) and (6) remain valid also by using a filtered version of
the signals fi (t), i = 1, ..., 4. Such an idea is used in [1] to estimate the time
delay along with other model parameters in an iterative way from an informative data set. In this paper we use an integrator as filter, because it allows
to introduce at least an integral effect on each term which contains the signal
y(t), i.e. the corrupting high-frequency noises on the signal are attenuated.
5
Remark 2 High frequency zero mean disturbances on the process output are
filtered by the integration operations, so that their contribution to fi (t), i =
1, ..., 4 is negligible. Low-frequency noise and offset errors could cause estimation errors to the proposed method. This is a common problem to any identification method using step tests. The test signal should enable us to inject as
much energy as possible, or a high signal-to-noise ratio is required [9]. This implies that offsets should be small. If there exist inherent offsets, they may cause
significant estimation error in K, which further leads to estimation errors in
the other parameters.
Recalling that θ = LT , the relative error between the estimated values T̂ , L̂
and θ̂:
θ̂
(16)
ê = − 1 ,
T̂ L̂
can be used to check the consistency of the data with the model (1). For
example, given a threshold E, then a true/false criterion can be defined as:


 true, ê < E,
model hypothesis = 
(17)
 false, otherwise.
Remark 3 If the aim is to have a criterion for the consistency of the data
with the model, then it is not necessary to know the response amplitude in
advance.
A modified-area method can be easily implemented, in which K̂ and T̂ar =
T̂ + L̂ are the estimated parameters by the previous approach and a refined
estimate T ∗ and L∗ is computed as
ef2 (T̂ar )
,
U0 K̂
(18)
L∗ = T̂ar − T ∗ .
(19)
T∗ =
3
Simulation examples
In this section the proposed step identification method is applied to several
typical processes. Without loss of generality, a unit step (U0 = 1) is employed
in all the simulations. Since the estimator error should be small in both time
and frequency domain, the time-domain identification error is measured by
the standard deviation:
X
1 n−1
[y(kTs ) − ŷ(kTs )]2 ,
1 =
n k=0
6
(20)
where y(kTs ) is the actual process output under a step change, while ŷ(kTs )
is the response of the estimated process under the same step change. The
frequency-domain identification error is measured by the worst-case error:

  Ĝ(iω) − G(iω) 
2 = max 
ω∈[0,ωu ]
|G(iω)|
× 100,
(21)

for frequencies from zero to the ultimate frequency of the real process, ωu .
The results of the proposed method (EF) are compared to those obtained by
the area method (AM) (in the noise-free cases) and the method in [3], (BI).
In all experiments we will use n = 10000 samples with an observation time
equal to Tobs = 10s, 80s, 30s, 30s for the four considered systems, respectively.
Example 1. Consider a first-order plus dead-time process
G(s) =
1 −s
e .
s+1
A unit step is performed, and the process output in the step response is
recorded from t = 0 to t = 10s. In the noise-free case, the model estimated
with the proposed method is
ĜEF =
1.0000
e−1.0000s .
1.0000s + 1
The identification errors are 1 = 9.72e−17 and 2 = 8.62e−06%. AM method
gives
0.9999
ĜAM =
e−1.0015s ,
0.9979s + 1
with 1 = 5.64e − 08 and 2 = 0.2711%, and BI method gives
ĜBI =
1.0000
e−1.0010s ,
1.0000s + 1
with 1 = 4.99e − 08 and 2 = 0.2029%.
Tests are also performed on other processes with different dynamics and the
results are summarized in Table 1. To show the robustness of the proposed
method the step tests are performed adding to the plant output a discrete
zero mean white noise with several value of the signal-to-noise ratio
v
u n−1
uP
2
u
u i=0 y(iTc )
u
.
SN R = u n−1
t P
2
i=0
7
r(iTc )
(22)
Example 2. Consider a 2nd-order plus dead-time process
G(s) =
e−4s
.
(10s + 1)(2s + 1)
Under measurement noise with SN R = 5, a step test is performed and the
process output is recorded from t = 0 to t = 80s. Since the estimated values
of the parameters using a single noise sequence are not true representative of
the performance of a parameter estimation method, we have generated 100
noise sequences to be summed to the output response and we save the mean
value of the estimated parameters and the errors 1 and 2 . The estimation
errors are 1 = 2.95e − 02 and 2 = 16.17%. Tables 2, 3 and 4 show the results
obtained with the other processes, for SN R = 100, 10, 5 respectively.
4
Conclusions
In this paper a method to identify first-order plus time-delay has been presented. It is based on a linear regression equation which involves only the
time constant and the delay. Noise sensitivity is avoided by filtered equations
based on integrations and convolutions operations of the output signal during the observation time. The method does not require complex numerical
calculations.
References
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continuous-time models using a linear filter”. Journal of Process Control, 16,
pp. 323–331, 2006.
[2] K. J. Aström and T. Hägglund, PID controllers: theory, design and tuning (2nd
ed.). NC, USA: Instrument Society of America, 1995.
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identification of first-order plus dead-time model from step response”. Control
Eng. Practice, 7, pp. 71–77, 1999.
[4] M. Fliess and H. Sira-Ramirez, “An algebraic framework for linear
identification”. ESAIM: Control, Optimisation and Calculus of Variations, 9,
pp. 151–168, 2003.
[5] A. Ingimundarson, and T. Hägglund, T., Closed-loop identification of a first-order
plus dead-time model with method of moments. IFAC, Int. Symp. on Advanced
Control of Chem. Proc., 2000, Pisa, Italy.
8
[6] S. Vivek and M. Chidambaram, “Identification using single symmetrical relay
feedback test”. Comp. & Chemical Eng., 29, pp. 1625–1630, 2005.
[7] L. Wang, M. L. Desarmo and W. R. Cluett, “Real-time estimation of process
frequency response and step response from relay feedback experiments”.
Automatica, 35, pp. 1427–1436, 1999.
[8] Q. G. Wang and Y. Zhang, “Robust identification of continuous system with
dead-time from step responses”. Automatica, 37, pp. 377–390, 2001.
[9] Q. G. Wang, G. Xin and Y. Zhang, “Direct identification of continuous time
delay systems from step responses”. J. Proc. Control, 11, pp. 531–542, 2001.
[10] T. Soderstrom and P.G. Stoica, System identification. New York: Prentice-Hall,
1989.
[11] T. Soderstrom and P.G. Stoica, Instrumental variable methods for system
identification. Berlin: Springer,1983.
9
e−s
s+1
e−4s
(10s+1)(2s+1)
1−s
(s+1)5
1
(s+1)8
Ĝ(s)
1.0000
−1.0000s
1.0000s+1 e
1.0015
−5.6474s
10.4495s+1 e
1.0010
−3.9232s
2.1005s+1 e
1.0110
−4.7421s
3.5007s+1 e
1
9.72e − 17
4.12e − 05
6.51e − 04
8.90e − 04
2
8.62e − 06%
14.93%
13.22%
23.30%
Ĝ(s)
0.9999
−1.0015s
0.9979s+1 e
0.9994
−5.8059s
10.1517s+1 e
1.0000
−3.8198s
2.1817s+1 e
1.0000
−4.9700s
3.0315s+1 e
1
5.64e − 08
5.58e − 05
5.48e − 04
1.20 − 03
2
0.27%
17.50%
11.21%
26.47%
Ĝ(s)
1.0000
−1.0010s
1.0000s+1 e
1.0008
−5.8503s
10.2057s+1 e
1.0013
−4.0456s
1.9826s+1 e
1.0120
−4.9155s
3.3161s+1 e
1
4.99e − 08
5.38e − 05
8.60e − 04
8.78 − 04
2
0.20%
17.35%
17.85%
21.22%
G(s)
EF
AM
BI
Table 1
Identification results without noise.
e−s
s+1
e−4s
(10s+1)(2s+1)
1−s
(s+1)5
1
(s+1)8
Ĝ(s)
1.0000
−0.9999s
1.0002s+1 e
1.0015
−5.6473s
10.4503s+1 e
1.0010
−3.9232s
2.1001s+1 e
1.0110
−4.7422s
3.5009s+1 e
1
7.50e − 05
1.15e − 04
7.28e − 04
9.58e − 04
2
0.20%
14.93%
13.23%
23.29%
Ĝ(s)
1.0000
−1.0010s
1.0000s+1 e
1.0008
−5.8508s
10.2059s+1 e
1.0013
−4.0453s
1.9826s+1 e
1.0120
−4.9157s
3.3163s+1 e
1
7.51e − 05
1.27e − 04
9.36e − 04
9.46e − 04
2
0.38%
17.35%
17.84%
21.21%
G(s)
EF
BI
Table 2
Identification results with SN R = 100.
e−s
s+1
e−4s
(10s+1)(2s+1)
1−s
(s+1)5
1
(s+1)8
Ĝ(s)
0.9999
−1.0012s
0.9985s+1 e
1.0014
−5.6414s
10.4483s+1 e
1.0011
−3.9178s
2.1027s+1 e
1.0111
−4.7441s
3.5016s+1 e
1
7.50e − 03
7.40e − 03
8.30e − 03
7.70e − 03
2
1.98%
15.28%
13.27%
23.27%
Ĝ(s)
0.9999
−1.0013s
0.9995s+1 e
1.0008
−5.8479s
10.2005s+1 e
1.0014
−4.0402s
1.9847s+1 e
1.0120
−4.9169s
3.3176s+1 e
1
7.50e − 03
7.40e − 03
8.50e − 03
7.70e − 03
2
1.86%
17.50%
17.73%
21.14%
G(s)
EF
BI
Table 3
Identification results with SN R = 10.
10
e−s
s+1
e−4s
(10s+1)(2s+1)
1−s
(s+1)5
1
(s+1)8
Ĝ(s)
0.9999
−0.9997s
1.0030s+1 e
1.0014
−5.6667s
10.4313s+1 e
1.0008
−3.9311s
2.0973s+1 e
1.0111
−4.7335s
3.5071s+1 e
1
3.00e − 02
2.95e − 02
3.12e − 02
2.81e − 02
2
4.69%
16.17%
13.87%
23.77%
Ĝ(s)
0.9998
−1.0015s
1.0015s+1 e
1.0008
−5.8665s
10.1937s+1 e
1.0012
−4.0489s
1.9839s+1 e
1.0119
−4.9133s
3.3135s+1 e
1
3.00e − 02
2.95e − 02
3.14e − 02
2.81e − 02
2
4.15%
18.12%
17.98%
21.42%
G(s)
EF
BI
Table 4
Identification results with SN R = 5.
11