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Research Article
Research on Identification Algorithm of Cascade Control System
Damin Ding ,
1
,
2
Yagang Wang ,
1
,
2
Wei Zhang ,
1
and Wei Chu
1
,
2
1
School of Optical-Electrical and Computer Engineering, University of Shanghai for Science and Technology,
Shanghai 200093, China
2
Rehabilitation Engineering and Technology Institute, University of Shanghai for Science and Technology,
Shanghai 200093, China
Correspondence should be addressed to Yagang Wang; ygwang@usst.edu.cn
Received 4 June 2022; Revised 25 July 2022; Accepted 26 September 2022; Published 24 November 2022
Academic Editor: Ricardo Aguilar-Lopez
Copyright ©2022 Damin Ding et al. is is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Aiming at the problem of object model identication of modern industrial process control systems, a new closed-loop moment
parameter identication online method based on the data of normal operation of the running system is proposed. In this method,
only one step response data of the system is required, and appropriate convergence factors are introduced into the Laplace
formula, the trapezoidal integral method is used to calculate the values of two derivatives of the transfer function, then the four
unknown parameters of the second-order model can be solved by tting the data with the least square method, and the target
model can be identied. Finally, the simulation results of building dierent objects through Matlab show that the identication
method has general applicability and good robustness with high recognition, and it is not sensitive to noise signals.
1. Introduction
In the actual industrial production environment, the internal
structure of the object model is always equal to a black box.
We do not know the internal structure, and it is not simple to
restore the internal modeling to get an accurate model. e
idea of system identication is to obtain a model equivalent
to the internal model.
Arif et al. provided a thorough survey on the academic
research progress and industry practices and highlighted
existing issues and new trends in load modeling [1].
Podvalny and Vasiljev oered the stage-by-stage compo-
sition and identication of the non-linear model param-
eters that are based on the principles of a multi-alternative
structure and functioning of composite systems, i.e., multi-
levelness, modularity, and separation of functions, and that
method can overcome the basic restrictions caused by
incommutativity of non-linear and linear nodes of a system
[2]. A non-linear cascaded system identication approach
is presented in [3] to predict the model structure param-
eters that minimize the dierence between the estimated
and measured data, using benchmark datasets. Zhang et al.
proposed an eective control loop performance assessment
(CPA) of a cascade control system for many non-Gaussian
distributions even the unknown mixture disturbance noise
[4]. Kien and Anh proposed a new cascade training multi-
layer fuzzy logic identifying the forward model of double-
coupled tank system based on experiment [5]. Mavkov et al.
introduced a neural network architecture, called integrated
neural network (INN), for the identication of non-linear
continuous-time dynamical models in state-space repre-
sentation [6]. Kien et al. proposed a new cascade training
multi-layer fuzzy logic for identifying forward model of
multiple-input multiple-output (MIMO) non-linear dou-
ble-coupled uid tank system based on experiment plat-
form [7], and the cascade training using optimization
algorithms optimally trained multi-layer fuzzy model one
by one. All parameters of the multi-layer fuzzy model were
optimally and comparatively identied using DE, GA, and
PSO algorithms. Khorasani and Weyer [8] extended the
SPS approach to EIV systems to construct a condence
region for a single module in a simple cascade network by
incorporating additional data and taking advantage of the
cascade structure without estimating other modules in the
network. V¨
or¨
os [9] used three-block cascade models with
non-linear static, linear dynamic, and non-linear dynamic
Hindawi
Mathematical Problems in Engineering
Volume 2022, Article ID 3997081, 13 pages
https://doi.org/10.1155/2022/3997081
blocks to deal with the recursive identication of time-
varying non-linear dynamic systems. Mattsson et al. pro-
posed an identication method that uses a likelihood-based
methodology that adaptively penalizes model complexity
and directly leads to a computationally ecient imple-
mentation [10]. A hybrid backtracking search algorithm
with wavelet mutation (BSA-WM) has been applied for the
identication of the parameters associated with a non-
linear three-stage cascaded Wiener–Hammerstein (W-H)
system with polynomial non-linearities of dierent order
[11]. Karachalios et al. [12] presented an algorithm for data-
driven identication of non-linear cascaded systems with
Hammerstein structure relying on the Loewner framework
(LF). V¨
or¨
os [13] presented a new approach to the pa-
rameter identication of non-linear dynamic systems using
cascade models with non-linear dynamic, linear dynamic,
and non-linear static blocks based on a least-squares it-
erative technique. Most of the system identication algo-
rithms given in these studies are oine system
identication based on the acquired data [14–16]. When
encountering an unfamiliar controlled object system, these
system identication algorithms require the controlled
object to be shut down or out of the production process and
stimulated with a step input signal more than once. e
data generated by the controlled object are used for system
identication. In engineering applications, we often en-
counter situations where we need to adjust the parameters
of an unfamiliar controlled object. Sometimes, we do not
have enough time and nancial resources to obtain re-
sponse data through multiple trials.
e specic categories of model identication methods
can be divided into the identication of target objects
through the FOPDT (rst-order plus dead time) model and
the SOPDT (second-order plus dead time) model. It is often
necessary to rely on some methods of minimizing the error
criterion. We use the least-squares method, which is rela-
tively mature, has fewer iterations, and has better robustness.
In addition to the least-squares method, there are Newton
iteration method, particle swarm method, auxiliary variable
method, etc. [17–19], but the above methods have dierent
defects and are not suitable for practical applications. For
example, Newton’s iteration method [20, 21], duelist algo-
rithm [22], bargaining game theory-based approach [23],
biquad lters [24], and other similar algorithms [25] are too
computationally expensive, particle swarm method itera-
tions are too many, and so on. Normally, system identi-
cation is for the identication of a closed-loop system in an
open-loop situation. However, in some cases, the closed-
loop system does not allow the feedback loop to be dis-
connected [26].
Some recent studies such as Aljamaan et al. [3], Kien
et al. [5, 7], V¨
or¨
os [9], Mattsson et al. [10], and so on used the
oine system response data that have been obtained for
system identication. And these earlier studies of system
identication have been improved by Li et al. [27], Liu and
Gao [28] and Wang et al. [29]. Branko [30], Nino et al. [31],
and Chao et al. [32]’s methods seem to be suitable for only a
few scenarios and are not suitable for generalizing to other
areas.
Aiming at these problems, a closed-loop moment
identication method based on the data of the normal
operation of the online system is proposed.
e contributions of this paper can be summarized as
follows:
(1) A new closed-loop moment parameter identication
online method based on the data of normal opera-
tion of the running system is proposed, which only
needs to obtain the normal operating data of the
system setting value change once.
(2) We introduce a suitable convergence factor λinto the
Laplace formula as the best parameter to make the
signal converge quickly.
(3) We propose a method of trapezoidal integration to
calculate the value of the two derivations of the
transfer function to solve the four unknown pa-
rameters of the second-order model by the least-
squares method.
is paper selects dierent types of controlled objects for
step response simulation, such as simple control system,
non-minimum phase control system, and cascade control
system. According to the analysis of input and output sig-
nals, the transfer function in the frequency domain is dif-
ferentiated, and the signal data are processed by trapezoidal
integration. According to the algorithm in this paper, the
relevant parameters of the transfer function model can be
obtained. Judge the degree of t between the identication
model and the original model through the model’s Nyquist
graph and the output error value.
e rest of this paper is organized as follows. Section 1
provides the cascade control system signal analysis, and
object model identication is presented in Section 2. e
simulations and evaluations are presented in Section 3.
Finally, we present our conclusion in Section 5.
2. Signal Analysis
e magnetic heating stirrer is a laboratory instrument used
for liquid mixing, mainly for heating and stirring low vis-
cosity liquids or solid-liquid mixtures. e basic principle of
this stirrer is that the magnetic eld repels and attracts
opposites. It drives the magnetic stirrer placed in the vessel
to operate in a circular motion by changing the magnetic
eld, thus achieving the purpose of stirring the liquid. ere
are many types of magnetic stirrers. In addition to the
stirring function, it also has additional functions such as
heating temperature control and timing. e main types are
magnetic stirrer, magnetic heating stirrer, and constant
temperature heating stirrer. It has a heating temperature
control system, which can heat and control the sample
temperature according to the specic experimental re-
quirements, maintain the temperature conditions required
for the experimental conditions, and ensure that the liquid
mixing meets the experimental requirements.
e goal of this paper is to convert the system shown in
Figure 1 into an accurate mathematical model, thus facili-
tating the tuning of the controller. e system shown in
2Mathematical Problems in Engineering
Figure 1 is separated into internal and external loops, and the
system models are identied separately based on normal
operation data.
Since the internal loop is usually a non-minimum phase
system, we need to verify the applicability of the method in
this paper to the non-minimum phase system rst and then
to the overall cascade system.
In this paper, the temperature is considered the con-
trolled variable. In this system as shown in Figure 1, the
internal heater temperature is considered as the original
signal, i.e., the desired value. e heater tray temperature is
considered as the internal loop G2 output value and the
water temperature is considered as the external loop G1
output value. e heater acts on the system tray, and the
signal from the tray temperature sensor is treated as the
internal loop output, which is treated as the external loop
input, and the actual water temperature is treated as the
external loop output so that it can be identied and con-
verted into the mathematical model problem shown in
Figure 2 in this paper.
e model of the stirrer is shown in Figure 2, where u(s)
is the input signal, eext(s)is the outer loop noise, and eint(s)
is the inner loop noise. A convergence factor is introduced
here to avoid the integrand function not being able to be
integrated, and it can converge the response signal after the
signal reaches a steady state.
3. Object Model Identification
3.1. Identication Method Based on the Non-Minimum Phase
Systems. A non-minimum phase system refers to a con-
trolled object whose object transfer function has zeros, poles,
or a time-delay process on the right half of the complex
plane. Such systems with non-minimum phase objects are
Heating plate
temperature
Wat e r
temperature
G1
G2
(a) (b)
Figure 1: A typical stirrer with heating function. (a) Schematic diagram. (b) Real photos.
Mathematical Problems in Engineering 3
widely used in chemical industry reaction process con-
trollers, power electronic converters, automatic underwater
navigation controllers, and so on. Among the many types,
the non-minimum phase object with the right half complex
plane zero and the time-delay link is the most typical. When
the controlled object has the above characteristics at the
same time, it is a considerable challenge for the overall
system tuning and control. erefore, it is particularly
important to be able to accurately identify the target object
model.
In the actual industrial control object, due to the
complexity of its order, it is more dicult to design the
controller. erefore, a second-order model with a non-
minimum phase is proposed to t and approximate the
actual target system. Its form is as follows:
G(S) � k−s
as2+bs +ce−Ls.(1)
Put s�jωinto (1):
|G(jω)|∗arg[G(jω)] � k−jω
a(jω)2+b(jω) + ce−Ljω.(2)
Split (2) into amplitude and phase, as follows:
|G(jω)|2c−aω2
2+b2
�k2+ω2,(3)
ωL� −arg[G(jω)] − tan−1ω
k
−tan−1bω
c−aω2
.(4)
When the frequency value takes dierent values, both (3)
and (4) are written in the form of matrix multiplication:
Ax �B, (5)
c2
k2|G(jω)|2+b2−2ac
k2ω2|G(jω)|2+a2
k2ω4|G(jω)|2−1
k2ω2�1,(6)
x�
x1
x2
x3
x4
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
�
c2
k2
b2−2ac
k2
a2
k2
1
k2
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
,
(7)
B�111... 1
T.(8)
G2(s)
u (s)
eext(s) eint(s)
G1(s)
Figure 2: Mathematical model of the plant. e heater acts on the system tray, and the signal from the tray temperature sensor is treated as
the internal loop output, which is treated as the external loop input, and the actual water temperature is treated as the external loop output so
that it can be identied and converted into the mathematical model problem.
4Mathematical Problems in Engineering
xin equation (11) can be obtained by using the least-
squares method to solve, and the unknown parameter is
obtained by further solving:
x�AT·A
−1·AT·B, (9)
k
a
b
c
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦�
��
1
x4
�����������
x2+2����
x1x3
√
x4
��
x1
x4
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
.(10)
3.2. Identication Method Based on the Industrial Cascade
System. As shown in the cascade control system shown in
Figure 3, SP1 is the target value set by the user, PV2 is the
output value of the internal loop object G2, PV1 is the output
value of the external loop object G1, CO2 is the internal loop
PID controller GC2 output value, and SP2 is the output of
external loop PID controller GC1. G1 and G2 are the transfer
functions of the internal and external ring objects that need
to be identied, respectively.
In the cascade system, two models require model
identication, namely, G1 and G2. By setting the input value
of SP1, the input and output data of the two object models
are obtained, respectively, and the data can be obtained by
dividing by the Laplace transform. Equation (16) is a sim-
plied expression for the cascading system. e detailed
derivation process of this identication method will be
described below.
G1(s) � PV1(s)
CO1(s),(11)
G2(s) � PV2(s)
CO2(s).(12)
In this section, the second-order hysteresis model is used
to identify the internal and external objects in the cascade
control system, and simulation verication is given. e
results show that the method has good accuracy.
e model of this paper is
G(s) � q
as2+bs +1e−Ls.(13)
By collecting the signals at both ends of the internal loop
and external loop of the cascade control system, the cal-
culation formula is
Y(s) � ∞
0
y(t)e−stdt, (14)
R(s) � ∞
0
r(t)e−stdt. (15)
Calculating the n-th order derivative on both ends of (14)
and (15), (16) and (17) are obtained:
Y(n)(s) � (−1)(n)∞
0
tne−sty(t)dt, (16)
R(n)(s) � (−1)(n)∞
0
tne−str(t)dt. (17)
e transfer function is transformed to the following
formulas for the rst and second order derivatives:
Y′(s) � G′(s)R(s) + G(s)R′(s),(18)
Y″(s) � G″(s)R(s) + 2G′(s)R′(s) + G(s)R″(s).(19)
Transform and bring convergence factor λinto (20) and
(21):
G′(λ) � Y′(λ) − G(λ)R′(λ)
R(λ),(20)
G″(λ) � Y′(λ) − 2G′(λ)R′(λ) − G(λ)R″(λ)
R(λ).(21)
In (20) and (21), G(λ),G′(λ), and G″(λ)are all obtained
by obtaining the data trapezoidal integral (19) of the re-
sponse of the object. In order to reduce the amount of
calculation, take the logarithm of both ends of equation (25)
to get
lnG(s)�lnq−ln as2+bs +1
−Ls. (22)
Take the derivative of the two ends of (20) twice and
denote it as J1(s)and J2(s).
G′(s)
G(s)� − 2as +b
as2+bs +1−L⇔J1(s),(23)
G″(s)G(s) − G2(s)
G2(s)�2a2s2+b2+2abs −2a
as2+bs +1
2⇔J2(s).(24)
Transform (23) and bring s�λinto the following :
J2(λ) � −2λ2J2(λ) − 2
a−2λJ2(λ)b
+2λ2−λ4J2(λ)
a2+1−λ2J2(λ)
b2
+2λ−λ3J2(λ)
ab.
(25)
Let convergence factor λin the above formula take
dierent values:
λ�
λ1
λ2
λ3
···
λn
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
.(26)
Transform the (24) as follows, and use a matrix to ex-
press the following formula:
Mathematical Problems in Engineering 5
Bx �J2.(27)e matrix Bcorresponds to the coecient matrix of
(27), as follows:
B�
−2λ2
1J2λ1
−2−2λ1J2λ1
2λ2
1−λ4
1J2λ1
1−λ2
1J2λ1
2λ1−λ3
1J2λ1
−2λ2
2J2λ2
−2−2λ2J2λ2
2λ2
2−λ4
2J2λ2
1−λ2
2J2λ2
2λ2−λ3
2J2λ2
··· ··· ··· ··· ···
−2λ2
nJ2λn
−2−2λnJ2λn
2λ2
n−λ4
nJ2λn
1−λ2
nJ2λn
2λn−λ3
nJ2λn
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.(28)
e matrix J2corresponds to the value obtained by
taking dierent λvalues:
J2�
J2λ1
J2λ2
J2λ3
···
J2λn
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
.(29)
Unknown parameter matrix x:
x�
x1
x2
x3
x4
x5
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
�
a
b
a2
b2
ab
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
.
(30)
is paper uses the least-squares method to solve the
value of a,b:
x�BT·B
−1·BT·J2.(31)
According to the above model, the parameters can be
solved:
x1
x2
�a
b
.(32)
Incorporating the unknown parameter values, we get
L� − 2aλ+b
aλ2+bλ+1−J1(λ),(33)
q�G(λ)aλ2+bλ+1
e−λL.(34)
After many simulation experiments, it is found that the
value of λis preferably λ�0.16 +0.009nand nis between 5
and 15. e results show that after the parameters of the
second-order model with lag in (13) are returned, there is a
better identication eect.
4. Simulation
In order to verify the robustness and accuracy of the
identication method, simulation experiments were per-
formed on the two systems, respectively, and the verication
was veried by observing the target object model, the
Nyquist curve, and the error value of the identication
model of each method. In addition, a horizontal comparison
between the results of the method and the results of the
traditional rst-order identication algorithm is used to
verify the accuracy of the method. Interference noise is
added to the input and output signals to verify its anti-in-
terference ability and robustness. e concept of signal-to-
noise ratio (SNR) is introduced here, and the general noise
intensity is expressed by
NSR �mean(abs(noise))
mean(abs(signal)).(35)
NSR is the ratio of the average of the absolute values of all
noise signals to the average of the absolute values of the true
signals. e SNR algorithm is expressed as
SNR �20ln 1
NSR
(dB).(36)
In order to show the accuracy of the identication model
more intuitively, an error algorithm is introduced to evaluate
the accuracy of the model. e algorithm of εis shown in
SP1 SP2 CO2 PV2 CO1 PV1
GC1 GC2 G2 G1
Figure 3: Cascade control system. e control block diagram of the cascading system consists of two loops, the internal and external loop.
6Mathematical Problems in Engineering
ε�1
N
N
k�1[y(kT) − y(kT)]2.(37)
Among them, Nrepresents the sampling time of the step
response output data; Trepresents the sampling time; y(kT)
represents the output value of the k-th sampling point of the
identication model; and y(kT)represents the k-th sam-
pling point of the original model output value.
4.1. Non-Minimum Phase System Experiment. In this sec-
tion, build a simulation model for the non-minimum phase
system as shown in Figure 4.
For this model, the specic model is shown in
G(s) � −3s+2
2s2+3s+1e−5s.(38)
Using the frequency-domain identication method in this
article, under the noise interference of NSR �20%, the im-
portant frequency range obtained by data analysis is between
0 and 0.3450, and 30 sample data points are also taken in this
interval, and the nal object model is determined by tting
these data points as (39), and the nal object model is
compared with the results of the traditional methods
G(s) � −s+0.6667
0.6667s2+s+0.3333e−5s.(39)
e traditional method identication model is
G(s) � 1.992
11.5512s2+4.7563s+1e−3s.(40)
e simulation results of the models are compared in-
tuitively in the form of pictures as shown in Figure 5 and
Table 1. It can be seen from Figure 5 that the curve obtained
by the method in this paper can well t the curve of the
original plant. e curve obtained by the traditional method
is not very ideal.
It can be seen that the method in this paper is not weaker
than the traditional method from the comparison of the
Nyquist curves in Figure 6.
Figure 7 shows the results of the simulation mode of
Figure 4, where the three parameters of the PID are kp�
0.2186, ki�0.0423, and kd� −0.6026.
4.2. Industrial Cascade System Experiment. Similar to the
previous section, we should build a simulation model for the
cascade system as shown in Figure 8.
is model uses the industrial steam boiler object as the
object model. e specic model is shown in (41) and (42).
G(s) � 8
225s2+30s+1,(41)
G(s) � 1.125
(15s+1)3.(42)
e identication results of the method in this paper are
as follows: the internal ring model is shown in equation (43)
and the external ring model is shown in (44):
G(s) � 8.0003
223.9861s2+30.7805s+1e−0.2564s,(43)
G(s) � 1.1204
1050.7864s2+60.6804s+1e−10.0234s.(44)
Using the experimental data collected in the simulation
experiment obtained, the internal and external loop models
are identied by using the traditional method and the
method in this paper, respectively.
e results of the traditional method are as follows. e
internal loop model is shown in equation (45) and the
external loop model is shown in (46):
G(s) � 7.9795
236.459s2+31.0928s+1e−0.2891s,(45)
G(s) � 1.3085
633.277s2+51.095s+1e−0.6346s.(46)
It can be seen intuitively from the last two columns of
Table 2 that the direct error value between the recognition
model and the target model is less than 10e-04, which is 2
orders of magnitude smaller than the error value recognized
by the traditional method, which shows the accuracy of the
recognition model of this method. e performance indi-
cates that the method has high accuracy in identifying the
model.
It can be seen from Figures 9(a) and 9(b) that the curve of
internal loop and external loop obtained by the method in
this paper can well t the curve of the original plant. e
curve obtained by the traditional method is not very ideal.
It can be seen that the method in this paper is better than
the traditional method from the comparison of the Nyquist
curves in Figure 10.
Figure 11 shows the results of the simulation mode of
Figure 8, where the three parameters of the internal PID are
kp�0.0603, ki�4.5275, and kd� −4.2519, and the three
parameters of the external PID are kp�3.8293, ki�0.0206,
and kd� −834.1573. e selection of the control method
and the tuning of the control parameters will be presented in
the following work.
From the last two columns in Table 2 and Figure 10, it
can be seen intuitively that the direct error value between the
identication model and the target model is less than 10e−
04, which is two orders of magnitude smaller than the error
value identied by the traditional method. Order of mag-
nitude, which shows the high accuracy of the identication
model of this method, shows the high accuracy of the
identication model of this method.
e mean square error (MSE) of internal loop transfer
function and external loop transfer function obtained by
this method is compared with other studies [3, 33, 34] as
shown in Table 3. It is clear that the proposed CMP ap-
proach outperformed the other algorithms and has the
least MSE value. e algorithm CMP (closed-loop mo-
ment parameter identication) in this paper only needs to
obtain the normal operating data of the system setting
value change once, which means that the algorithm may
lead to overtting. ere are many studies [35–37]
Mathematical Problems in Engineering 7
1.1 1.2 1.3 1.4 1.5
−1.7
−1.6
−1.5
−1.4
−1.3
Real Axis
Object model
Method of this artical
Traditional method
Nyquist Diagram
Imaginary Axis
Figure 6: Comparison of Nyquist curves. ese are the Nyquist curves of the transfer function identied by the method of this paper and the
traditional method.
Table 1: Model of non-minimum phase system error value between the identication model and the target model.
Object model Method of this article Error Error for traditional method
G(s) � (−3s+2/2s2+3s+1)e−5sG(s) � (−s+0.6667/0.6667 s2+s+0.3333)e−5s1.2178e−08 1.3e−03
0 50 100 150 200 250 300
−10
0
10
20
30
40
Time (ms)
Original plant
Method in this paper
Traditional method
Figure 5: Comparison of the response curves of the three functions. e system identication of the original plant signal is done by the
traditional method and the method of this paper, respectively.
−
+PID (s) −3s + 2
2s2 + 3s + 1
Figure 4: e simulation mode of non-minimum phase system.
8Mathematical Problems in Engineering
−
+−
+
PID (s) PID (s) 8
225s2 + 30s + 1
1
15s + 1
1
15s + 1
1.125
15s + 1
Figure 8: e simulation of the cascade system.
0 50 100 150 200 250 300
−5
0
5
10
15
20
25
Time (ms)
Reference signal
Error
Control signal
Output
Figure 7: e simulation result of non-minimum phase system.
Table 2: Model of industrial cascade system experiment error value between the identication model and the target model.
Cascade system Object model Method of this article Error Error for traditional
method
Internal loop G(s) � (8/225s2+30s+1)G(s) � (8.0003/223.9861 s2+30.7805 s+1)e−0.2564 s5.7534e−07 1.2752
External loop G(s) � (1.125/(15s+1)3)G(s) � (1.1204/1050.7864 s2+60.6804 s+1)e−10.0234 s2.5913e−04 0.2377
0 200 400 600 800 1000
0
5
10
15
20
25
30
Time (ms)
Original plant
Method in this paper
Traditional method
(a)
Figure 9: Continued.
Mathematical Problems in Engineering 9
devoted to solving the problem of overtting. In Section 3,
two experiments were performed on the two systems,
respectively. Interference noise is added to the input and
output signals to verify its anti-interference ability and
robustness. Tables 1 and 2 show the noise results, which
show that the algorithm is insensitive to noise in system
identication. In the future, we will continue to study
online system identication, and under the framework of
Original plant
Method in this paper
Traditional method
0 200 400 600 800 1000
20
40
60
80
100
120
140
160
180
200
Time (ms)
(b)
Figure 9: Comparison of the response curves of the three functions. (a) e internal loop model. (b) e external loop model. is is a
comparison of the system response curves of the original plant signal, the transfer function identied by the method in this paper, and the
conventional method.
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8
−5.5
−5.0
−4.5
−4.0
−3.5
Real Axis
Object model
Method of this artical
Traditional method
Nyquist Diagram
Imaginary Axis
(a)
Object model
Method of this artical
Traditional method
0.6 0.7 0.8 0.9 1.0
−1.2
−1.1
−1.0
−0.9
−0.8
−0.7
−0.6
Real Axis
Nyquist Diagram
Imaginary Axis
(b)
Figure 10: Comparison of Nyquist curves of the internal and external loop model. (a) e internal loop model. (b) e external loop model.
is is a comparison of Nyquist curves of the original plant signal, the transfer function identied by the method in this paper, and the
conventional method.
10 Mathematical Problems in Engineering
the existing online system identication method pre-
sented in this paper, we will improve the method to
achieve more ideal results and nally apply it to engi-
neering practice.
5. Conclusion
is paper presents a model parameter identication method
for an industrial cascade system. e method does not require
any a priori condition and only needs to obtain the input and
output data under the step response to identify the controlled
objects, without complicated algorithm processing and a large
number of calculations. Simulations are performed for several
representative controlled objects, and the simulation results
show that the method identies the model with high accuracy
and strong anti-interference capability. Even in the case that
the signal-to-noise ratio is 20%, the recognition can be
completed well, and the recognition accuracy is very high,
which is better than some traditional recognition methods. e
higher the accuracy of the identied model of the controlled
object is, the better it is for the analysis of the controlled system
and the parameter setting of the controller, which facilitates the
whole control system to achieve a good control eect and can
be used to identify the actual industrial process object.
Data Availability
e Matlab code and Simulink data used to support the
ndings of this study have been deposited in the GitHub
repository (https://github.com/ericddm/
CascadeSystemIdentication-CMP).
Conflicts of Interest
e authors declare that there are no conicts of interest
regarding the publication of this paper.
Acknowledgments
is study was supported by the National Key Research and
Development Program of China (no. 2020YFC2007500).
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