# Research on Identification Algorithm of Cascade Control System

Authors: Damin Ding; Yagang Wang; Wei Zhang; Wei Chu
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Aiming at the problem of object model identification of modern industrial process control systems, a new closed-loop moment parameter identification 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 fitting the data with the least square method, and the target model can be identified. Finally, the simulation results of building different objects through Matlab show that the identification method has general applicability and good robustness with high recognition, and it is not sensitive to noise signals.


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## Body

## 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. The idea of system identification 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 offered the stage-by-stage composition and identification of the non-linear model parameters 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 identification approach is presented in [3] to predict the model structure parameters that minimize the difference between the estimated and measured data, using benchmark datasets. Zhang et al. proposed an effective 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 identification of non-linear continuous-time dynamical models in state-space representation [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 double-coupled fluid tank system based on experiment platform [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 identified using DE, GA, and PSO algorithms. Khorasani and Weyer [8] extended the SPS approach to EIV systems to construct a confidence 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. Vörös [9] used three-block cascade models with non-linear static, linear dynamic, and non-linear dynamic blocks to deal with the recursive identification of time-varying non-linear dynamic systems. Mattsson et al. proposed an identification method that uses a likelihood-based methodology that adaptively penalizes model complexity and directly leads to a computationally efficient implementation [10]. A hybrid backtracking search algorithm with wavelet mutation (BSA-WM) has been applied for the identification of the parameters associated with a non-linear three-stage cascaded Wiener–Hammerstein (W-H) system with polynomial non-linearities of different order [11]. Karachalios et al. [12] presented an algorithm for data-driven identification of non-linear cascaded systems with Hammerstein structure relying on the Loewner framework (LF). Vörös [13] presented a new approach to the parameter identification of non-linear dynamic systems using cascade models with non-linear dynamic, linear dynamic, and non-linear static blocks based on a least-squares iterative technique. Most of the system identification algorithms given in these studies are offline system identification based on the acquired data [14–16]. When encountering an unfamiliar controlled object system, these system identification 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. The data generated by the controlled object are used for system identification. In engineering applications, we often encounter situations where we need to adjust the parameters of an unfamiliar controlled object. Sometimes, we do not have enough time and financial resources to obtain response data through multiple trials.The specific categories of model identification methods can be divided into the identification of target objects through the FOPDT (first-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 relatively 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 different defects and are not suitable for practical applications. For example, Newton’s iteration method [20, 21], duelist algorithm [22], bargaining game theory-based approach [23], biquad filters [24], and other similar algorithms [25] are too computationally expensive, particle swarm method iterations are too many, and so on. Normally, system identification is for the identification 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 disconnected [26].Some recent studies such as Aljamaan et al. [3], Kien et al. [5, 7], Vörös [9], Mattsson et al. [10], and so on used the offline system response data that have been obtained for system identification. And these earlier studies of system identification 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 identification method based on the data of the normal operation of the online system is proposed.The contributions of this paper can be summarized as follows:(1)
A new closed-loop moment parameter identification online method based on the data of normal operation 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 factorλ into 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 parameters of the second-order model by the least-squares method.This paper selects different 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 signals, the transfer function in the frequency domain is differentiated, 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 fit between the identification model and the original model through the model’s Nyquist graph and the output error value.The rest of this paper is organized as follows. Section 1 provides the cascade control system signal analysis, and object model identification is presented in Section 2. The simulations and evaluations are presented in Section 3. Finally, we present our conclusion in Section 5.


## 2. Signal Analysis

The magnetic heating stirrer is a laboratory instrument used for liquid mixing, mainly for heating and stirring low viscosity liquids or solid-liquid mixtures. The basic principle of this stirrer is that the magnetic field repels and attracts opposites. It drives the magnetic stirrer placed in the vessel to operate in a circular motion by changing the magnetic field, thus achieving the purpose of stirring the liquid. There are many types of magnetic stirrers. In addition to the stirring function, it also has additional functions such as heating temperature control and timing. The 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 specific experimental requirements, maintain the temperature conditions required for the experimental conditions, and ensure that the liquid mixing meets the experimental requirements.The goal of this paper is to convert the system shown in Figure1 into an accurate mathematical model, thus facilitating the tuning of the controller. The system shown in Figure 1 is separated into internal and external loops, and the system models are identified separately based on normal operation data.Figure 1
A typical stirrer with heating function. (a) Schematic diagram. (b) Real photos.
(a)(b)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 first and then to the overall cascade system.In this paper, the temperature is considered the controlled variable. In this system as shown in Figure1, the internal heater temperature is considered as the original signal, i.e., the desired value. The 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. The 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 identified and converted into the mathematical model problem shown in Figure 2 in this paper.Figure 2
Mathematical model of the plant. The 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 identified and converted into the mathematical model problem.The model of the stirrer is shown in Figure2, where us is the input signal, eexts is the outer loop noise, and eints 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. Identification Method Based on the Non-Minimum Phase Systems

A non-minimum phase system refers to a controlled 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 widely used in chemical industry reaction process controllers, 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. Therefore, 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 difficult to design the controller. Therefore, a second-order model with a non-minimum phase is proposed to fit and approximate the actual target system. Its form is as follows:(1)GS=k−sas2+bs+ce−Ls.Puts=jω into (1):(2)Gjω∗argGjω=k−jωajω2+bjω+ce−Ljω.Split (2) into amplitude and phase, as follows:(3)Gjω2c−aω22+b2=k2+ω2,(4)ωL=−argGjω−tan−1ωk−tan−1bωc−aω2.When the frequency value takes different values, both (3) and (4) are written in the form of matrix multiplication:(5)Ax=B,(6)c2k2Gjω2+b2−2ack2ω2Gjω2+a2k2ω4Gjω2−1k2ω2=1,(7)x=x1x2x3x4=c2k2b2−2ack2a2k21k2,(8)B=111…1T.x in equation (11) can be obtained by using the least-squares method to solve, and the unknown parameter is obtained by further solving:(9)x=AT⋅A−1⋅AT⋅B,(10)kabc=1x4x2+2x1x3x4x1x4.


### 3.2. Identification Method Based on the Industrial Cascade System

As shown in the cascade control system shown in Figure3, 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 identified, respectively.Figure 3
Cascade control system. The control block diagram of the cascading system consists of two loops, the internal and external loop.In the cascade system, two models require model identification, 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 simplified expression for the cascading system. The detailed derivation process of this identification method will be described below.(11)G1s=PV1sCO1s,(12)G2s=PV2sCO2s.In this section, the second-order hysteresis model is used to identify the internal and external objects in the cascade control system, and simulation verification is given. The results show that the method has good accuracy.The model of this paper is(13)Gs=qas2+bs+1e−Ls.By collecting the signals at both ends of the internal loop and external loop of the cascade control system, the calculation formula is(14)Ys=∫0∞yte−stdt,(15)Rs=∫0∞rte−stdt.Calculating then-th order derivative on both ends of (14) and (15), (16) and (17) are obtained:(16)Yns=−1n∫0∞tne−stytdt,(17)Rns=−1n∫0∞tne−strtdt.The transfer function is transformed to the following formulas for the first and second order derivatives:(18)Y′s=G′sRs+GsR′s,(19)Y″s=G″sRs+2G′sR′s+GsR″s.Transform and bring convergence factorλ into (20) and (21):(20)G′λ=Y′λ−GλR′λRλ,(21)G″λ=Y′λ−2G′λR′λ−GλR″λRλ.In (20) and (21), Gλ, G′λ, and G″λ are all obtained by obtaining the data trapezoidal integral (19) of the response of the object. In order to reduce the amount of calculation, take the logarithm of both ends of equation (25) to get(22)lnGs=lnq−lnas2+bs+1−Ls.Take the derivative of the two ends of (20) twice and denote it as J1s and J2s.(23)G′sGs=−2as+bas2+bs+1−L⇔J1s,(24)G″sGs−G2sG2s=2a2s2+b2+2abs−2aas2+bs+12⇔J2s.Transform (23) and bring s=λ into the following :(25)J2λ=−2λ2J2λ−2a−2λJ2λb+2λ2−λ4J2λa2+1−λ2J2λb2+2λ−λ3J2λab.Let convergence factorλ in the above formula take different values:(26)λ=λ1λ2λ3⋯λn.Transform the (24) as follows, and use a matrix to express the following formula:(27)Bx=J2.The matrixB corresponds to the coefficient matrix of (27), as follows:(28)B=−2λ12J2λ1−2−2λ1J2λ12λ12−λ14J2λ11−λ12J2λ12λ1−λ13J2λ1−2λ22J2λ2−2−2λ2J2λ22λ22−λ24J2λ21−λ22J2λ22λ2−λ23J2λ2⋯⋯⋯⋯⋯−2λn2J2λn−2−2λnJ2λn2λn2−λn4J2λn1−λn2J2λn2λn−λn3J2λn.The matrixJ2 corresponds to the value obtained by taking different λ values:(29)J2=J2λ1J2λ2J2λ3⋯J2λn.Unknown parameter matrixx:(30)x=x1x2x3x4x5=aba2b2ab.This paper uses the least-squares method to solve the value ofa, b:(31)x=BT⋅B−1⋅BT⋅J2.According to the above model, the parameters can be solved:(32)x1x2=ab.Incorporating the unknown parameter values, we get(33)L=−2aλ+baλ2+bλ+1−J1λ,(34)q=Gλaλ2+bλ+1e−λL.After many simulation experiments, it is found that the value ofλ is preferably λ=0.16+0.009n and n is between 5 and 15. The results show that after the parameters of the second-order model with lag in (13) are returned, there is a better identification effect.


## 3.1. Identification Method Based on the Non-Minimum Phase Systems

A non-minimum phase system refers to a controlled 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 widely used in chemical industry reaction process controllers, 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. Therefore, 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 difficult to design the controller. Therefore, a second-order model with a non-minimum phase is proposed to fit and approximate the actual target system. Its form is as follows:(1)GS=k−sas2+bs+ce−Ls.Puts=jω into (1):(2)Gjω∗argGjω=k−jωajω2+bjω+ce−Ljω.Split (2) into amplitude and phase, as follows:(3)Gjω2c−aω22+b2=k2+ω2,(4)ωL=−argGjω−tan−1ωk−tan−1bωc−aω2.When the frequency value takes different values, both (3) and (4) are written in the form of matrix multiplication:(5)Ax=B,(6)c2k2Gjω2+b2−2ack2ω2Gjω2+a2k2ω4Gjω2−1k2ω2=1,(7)x=x1x2x3x4=c2k2b2−2ack2a2k21k2,(8)B=111…1T.x in equation (11) can be obtained by using the least-squares method to solve, and the unknown parameter is obtained by further solving:(9)x=AT⋅A−1⋅AT⋅B,(10)kabc=1x4x2+2x1x3x4x1x4.


## 3.2. Identification Method Based on the Industrial Cascade System

As shown in the cascade control system shown in Figure3, 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 identified, respectively.Figure 3
Cascade control system. The control block diagram of the cascading system consists of two loops, the internal and external loop.In the cascade system, two models require model identification, 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 simplified expression for the cascading system. The detailed derivation process of this identification method will be described below.(11)G1s=PV1sCO1s,(12)G2s=PV2sCO2s.In this section, the second-order hysteresis model is used to identify the internal and external objects in the cascade control system, and simulation verification is given. The results show that the method has good accuracy.The model of this paper is(13)Gs=qas2+bs+1e−Ls.By collecting the signals at both ends of the internal loop and external loop of the cascade control system, the calculation formula is(14)Ys=∫0∞yte−stdt,(15)Rs=∫0∞rte−stdt.Calculating then-th order derivative on both ends of (14) and (15), (16) and (17) are obtained:(16)Yns=−1n∫0∞tne−stytdt,(17)Rns=−1n∫0∞tne−strtdt.The transfer function is transformed to the following formulas for the first and second order derivatives:(18)Y′s=G′sRs+GsR′s,(19)Y″s=G″sRs+2G′sR′s+GsR″s.Transform and bring convergence factorλ into (20) and (21):(20)G′λ=Y′λ−GλR′λRλ,(21)G″λ=Y′λ−2G′λR′λ−GλR″λRλ.In (20) and (21), Gλ, G′λ, and G″λ are all obtained by obtaining the data trapezoidal integral (19) of the response of the object. In order to reduce the amount of calculation, take the logarithm of both ends of equation (25) to get(22)lnGs=lnq−lnas2+bs+1−Ls.Take the derivative of the two ends of (20) twice and denote it as J1s and J2s.(23)G′sGs=−2as+bas2+bs+1−L⇔J1s,(24)G″sGs−G2sG2s=2a2s2+b2+2abs−2aas2+bs+12⇔J2s.Transform (23) and bring s=λ into the following :(25)J2λ=−2λ2J2λ−2a−2λJ2λb+2λ2−λ4J2λa2+1−λ2J2λb2+2λ−λ3J2λab.Let convergence factorλ in the above formula take different values:(26)λ=λ1λ2λ3⋯λn.Transform the (24) as follows, and use a matrix to express the following formula:(27)Bx=J2.The matrixB corresponds to the coefficient matrix of (27), as follows:(28)B=−2λ12J2λ1−2−2λ1J2λ12λ12−λ14J2λ11−λ12J2λ12λ1−λ13J2λ1−2λ22J2λ2−2−2λ2J2λ22λ22−λ24J2λ21−λ22J2λ22λ2−λ23J2λ2⋯⋯⋯⋯⋯−2λn2J2λn−2−2λnJ2λn2λn2−λn4J2λn1−λn2J2λn2λn−λn3J2λn.The matrixJ2 corresponds to the value obtained by taking different λ values:(29)J2=J2λ1J2λ2J2λ3⋯J2λn.Unknown parameter matrixx:(30)x=x1x2x3x4x5=aba2b2ab.This paper uses the least-squares method to solve the value ofa, b:(31)x=BT⋅B−1⋅BT⋅J2.According to the above model, the parameters can be solved:(32)x1x2=ab.Incorporating the unknown parameter values, we get(33)L=−2aλ+baλ2+bλ+1−J1λ,(34)q=Gλaλ2+bλ+1e−λL.After many simulation experiments, it is found that the value ofλ is preferably λ=0.16+0.009n and n is between 5 and 15. The results show that after the parameters of the second-order model with lag in (13) are returned, there is a better identification effect.


## 4. Simulation

In order to verify the robustness and accuracy of the identification method, simulation experiments were performed on the two systems, respectively, and the verification was verified by observing the target object model, the Nyquist curve, and the error value of the identification model of each method. In addition, a horizontal comparison between the results of the method and the results of the traditional first-order identification algorithm is used to verify the accuracy of the method. Interference noise is added to the input and output signals to verify its anti-interference ability and robustness. The concept of signal-to-noise ratio (SNR) is introduced here, and the general noise intensity is expressed by(35)NSR=meanabsnoisemeanabssignal.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. The SNR algorithm is expressed as(36)SNR=20ln1NSRdB.In order to show the accuracy of the identification model more intuitively, an error algorithm is introduced to evaluate the accuracy of the model. The algorithm ofε is shown in(37)ε=1N∑k=1Ny^kT−ykT2.Among them,N represents the sampling time of the step response output data; T represents the sampling time; y^kT represents the output value of the k-th sampling point of the identification model; and ykT represents the k-th sampling point of the original model output value.


### 4.1. Non-Minimum Phase System Experiment

In this section, build a simulation model for the non-minimum phase system as shown in Figure4.Figure 4
The simulation mode of non-minimum phase system.For this model, the specific model is shown in(38)Gs=−3s+22s2+3s+1e−5s.Using the frequency-domain identification method in this article, under the noise interference ofNSR=20%, the important 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 final object model is determined by fitting these data points as (39), and the final object model is compared with the results of the traditional methods(39)Gs=−s+0.66670.6667s2+s+0.3333e−5s.The traditional method identification model is(40)Gs=1.99211.5512s2+4.7563s+1e−3s.The simulation results of the models are compared intuitively in the form of pictures as shown in Figure5 and Table 1. It can be seen from Figure 5 that the curve obtained by the method in this paper can well fit the curve of the original plant. The curve obtained by the traditional method is not very ideal.Figure 5
Comparison of the response curves of the three functions. The system identification of the original plant signal is done by the traditional method and the method of this paper, respectively.Table 1
Model of non-minimum phase system error value between the identification model and the target model.
Object modelMethod of this articleErrorError for traditional methodGs=−3s+2/2s2+3s+1e−5sGs=−s+0.6667/0.6667s2+s+0.3333e−5s1.2178e −081.3e −03It can be seen that the method in this paper is not weaker than the traditional method from the comparison of the Nyquist curves in Figure6.Figure 6
Comparison of Nyquist curves. These are the Nyquist curves of the transfer function identified by the method of this paper and the traditional method.Figure7 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.Figure 7
The simulation result of non-minimum phase system.


### 4.2. Industrial Cascade System Experiment

Similar to the previous section, we should build a simulation model for the cascade system as shown in Figure8.Figure 8
The simulation of the cascade system.This model uses the industrial steam boiler object as the object model. The specific model is shown in (41) and (42).(41)Gs=8225s2+30s+1,(42)Gs=1.12515s+13.The identification 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):(43)Gs=8.0003223.9861s2+30.7805s+1e−0.2564s,(44)Gs=1.12041050.7864s2+60.6804s+1e−10.0234s.Using the experimental data collected in the simulation experiment obtained, the internal and external loop models are identified by using the traditional method and the method in this paper, respectively.The results of the traditional method are as follows. The internal loop model is shown in equation (45) and the external loop model is shown in (46):(45)Gs=7.9795236.459s2+31.0928s+1e−0.2891s,(46)Gs=1.3085633.277s2+51.095s+1e−0.6346s.It can be seen intuitively from the last two columns of Table2 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. The performance indicates that the method has high accuracy in identifying the model.Table 2
Model of industrial cascade system experiment error value between the identification model and the target model.
Cascade systemObject modelMethod of this articleErrorError for traditional methodInternal loopGs=8/225s2+30s+1Gs=8.0003/223.9861s2+30.7805s+1e−0.2564s5.7534e −071.2752External loopGs=1.125/15s+13Gs=1.1204/1050.7864s2+60.6804s+1e−10.0234s2.5913e −040.2377It can be seen from Figures9(a) and 9(b) that the curve of internal loop and external loop obtained by the method in this paper can well fit the curve of the original plant. The curve obtained by the traditional method is not very ideal.Figure 9
Comparison of the response curves of the three functions. (a) The internal loop model. (b) The external loop model. This is a comparison of the system response curves of the original plant signal, the transfer function identified by the method in this paper, and the conventional method.
(a)(b)It can be seen that the method in this paper is better than the traditional method from the comparison of the Nyquist curves in Figure10.Figure 10
Comparison of Nyquist curves of the internal and external loop model. (a) The internal loop model. (b) The external loop model. This is a comparison of Nyquist curves of the original plant signal, the transfer function identified by the method in this paper, and the conventional method.
(a)(b)Figure11 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. The selection of the control method and the tuning of the control parameters will be presented in the following work.Figure 11
The simulation result of the cascade system. It shows the control signal and output signal of the inner loop and the output signal of the external loop, in which the output signal of the inner loop is the input signal of the external loop.From the last two columns in Table2 and Figure 10, it can be seen intuitively that the direct error value between the identification model and the target model is less than 10e−04, which is two orders of magnitude smaller than the error value identified by the traditional method. Order of magnitude, which shows the high accuracy of the identification model of this method, shows the high accuracy of the identification model of this method.The 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 approach outperformed the other algorithms and has the least MSE value. The algorithm CMP (closed-loop moment parameter identification) 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 overfitting. There are many studies [35–37] devoted to solving the problem of overfitting. 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 identification. In the future, we will continue to study online system identification, and under the framework of the existing online system identification method presented in this paper, we will improve the method to achieve more ideal results and finally apply it to engineering practice.Table 3
A comparison of MSE results of step response using different approaches.
MethodMSE resultPNLSS-I (polynomial non-linear state space)0.4498NLSS20.3433BLA (best linear approximation)0.7574Volterra models (3rd degree) (transient removal)0.5400Volterra models (3rd degree) (no transient est.)2.0800HBJ0.0087Proposed CMP (closed-loop moment parameter identification)0.0049


## 4.1. Non-Minimum Phase System Experiment

In this section, build a simulation model for the non-minimum phase system as shown in Figure4.Figure 4
The simulation mode of non-minimum phase system.For this model, the specific model is shown in(38)Gs=−3s+22s2+3s+1e−5s.Using the frequency-domain identification method in this article, under the noise interference ofNSR=20%, the important 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 final object model is determined by fitting these data points as (39), and the final object model is compared with the results of the traditional methods(39)Gs=−s+0.66670.6667s2+s+0.3333e−5s.The traditional method identification model is(40)Gs=1.99211.5512s2+4.7563s+1e−3s.The simulation results of the models are compared intuitively in the form of pictures as shown in Figure5 and Table 1. It can be seen from Figure 5 that the curve obtained by the method in this paper can well fit the curve of the original plant. The curve obtained by the traditional method is not very ideal.Figure 5
Comparison of the response curves of the three functions. The system identification of the original plant signal is done by the traditional method and the method of this paper, respectively.Table 1
Model of non-minimum phase system error value between the identification model and the target model.
Object modelMethod of this articleErrorError for traditional methodGs=−3s+2/2s2+3s+1e−5sGs=−s+0.6667/0.6667s2+s+0.3333e−5s1.2178e −081.3e −03It can be seen that the method in this paper is not weaker than the traditional method from the comparison of the Nyquist curves in Figure6.Figure 6
Comparison of Nyquist curves. These are the Nyquist curves of the transfer function identified by the method of this paper and the traditional method.Figure7 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.Figure 7
The simulation result of non-minimum phase system.


## 4.2. Industrial Cascade System Experiment

Similar to the previous section, we should build a simulation model for the cascade system as shown in Figure8.Figure 8
The simulation of the cascade system.This model uses the industrial steam boiler object as the object model. The specific model is shown in (41) and (42).(41)Gs=8225s2+30s+1,(42)Gs=1.12515s+13.The identification 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):(43)Gs=8.0003223.9861s2+30.7805s+1e−0.2564s,(44)Gs=1.12041050.7864s2+60.6804s+1e−10.0234s.Using the experimental data collected in the simulation experiment obtained, the internal and external loop models are identified by using the traditional method and the method in this paper, respectively.The results of the traditional method are as follows. The internal loop model is shown in equation (45) and the external loop model is shown in (46):(45)Gs=7.9795236.459s2+31.0928s+1e−0.2891s,(46)Gs=1.3085633.277s2+51.095s+1e−0.6346s.It can be seen intuitively from the last two columns of Table2 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. The performance indicates that the method has high accuracy in identifying the model.Table 2
Model of industrial cascade system experiment error value between the identification model and the target model.
Cascade systemObject modelMethod of this articleErrorError for traditional methodInternal loopGs=8/225s2+30s+1Gs=8.0003/223.9861s2+30.7805s+1e−0.2564s5.7534e −071.2752External loopGs=1.125/15s+13Gs=1.1204/1050.7864s2+60.6804s+1e−10.0234s2.5913e −040.2377It can be seen from Figures9(a) and 9(b) that the curve of internal loop and external loop obtained by the method in this paper can well fit the curve of the original plant. The curve obtained by the traditional method is not very ideal.Figure 9
Comparison of the response curves of the three functions. (a) The internal loop model. (b) The external loop model. This is a comparison of the system response curves of the original plant signal, the transfer function identified by the method in this paper, and the conventional method.
(a)(b)It can be seen that the method in this paper is better than the traditional method from the comparison of the Nyquist curves in Figure10.Figure 10
Comparison of Nyquist curves of the internal and external loop model. (a) The internal loop model. (b) The external loop model. This is a comparison of Nyquist curves of the original plant signal, the transfer function identified by the method in this paper, and the conventional method.
(a)(b)Figure11 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. The selection of the control method and the tuning of the control parameters will be presented in the following work.Figure 11
The simulation result of the cascade system. It shows the control signal and output signal of the inner loop and the output signal of the external loop, in which the output signal of the inner loop is the input signal of the external loop.From the last two columns in Table2 and Figure 10, it can be seen intuitively that the direct error value between the identification model and the target model is less than 10e−04, which is two orders of magnitude smaller than the error value identified by the traditional method. Order of magnitude, which shows the high accuracy of the identification model of this method, shows the high accuracy of the identification model of this method.The 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 approach outperformed the other algorithms and has the least MSE value. The algorithm CMP (closed-loop moment parameter identification) 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 overfitting. There are many studies [35–37] devoted to solving the problem of overfitting. 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 identification. In the future, we will continue to study online system identification, and under the framework of the existing online system identification method presented in this paper, we will improve the method to achieve more ideal results and finally apply it to engineering practice.Table 3
A comparison of MSE results of step response using different approaches.
MethodMSE resultPNLSS-I (polynomial non-linear state space)0.4498NLSS20.3433BLA (best linear approximation)0.7574Volterra models (3rd degree) (transient removal)0.5400Volterra models (3rd degree) (no transient est.)2.0800HBJ0.0087Proposed CMP (closed-loop moment parameter identification)0.0049


## 5. Conclusion

This paper presents a model parameter identification method for an industrial cascade system. The 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 identifies 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. The higher the accuracy of the identified 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 effect and can be used to identify the actual industrial process object.



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