# Welding Process Tracking Control Based on Multiple Model Iterative Learning Control

Authors: Xiaoli Li; Jian Liu; Linkun Wang; Kang Wang; Yang Li
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

The welding of the same parts has same welding trajectory, so welding process has strong repeatability. In this paper, aiming at the repeatability of welding process, an iterative learning controller is designed to achieve the control of weld quality. Due to the extremely variable welding environment and the presence of noise interferences and load disturbances, it is easy to cause the jumping change in parameters and even the structure of the welding system. Therefore, the idea of multiple model adaptive control (MMAC) is introduced into iterative learning control (ILC), and a multiple model iterative learning control (MMILC) algorithm is designed according to model of weld pool dynamic process in gas tungsten arc welding (GTAW). Besides, the convergence of the algorithm is analyzed for two cases: fixed parameters and jumping parameters. It turns out that the MMILC can not only utilize the repetitive information effectively in the welding process to achieve high precision tracking control of weld seam in limited time interval, but also realize the multiple model switching according to different working conditions to improve the welding quality.


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## Body

## 1. Introduction

Welding is an important process in modern industrial production. It is an operation method of joining two or more base metals into a whole body through welding materials under high temperature or high pressure conditions. As an important welding technology, gas tungsten arc welding (GTAW) is mainly used to weld easily oxidized alloys, stainless steel, and other refractory active metals, and it is also widely used in aerospace and many other industrial fields. During the welding process, an arc is generated between the tungsten electrode and the base material for welding, and at the same time, argon gas, which is used as a shielding gas to isolate air, is ejected continuously from welding torch muzzle to form a protective layer around the arc. In this way, high quality welds can be obtained. Nowadays, high quality and high efficiency are pursued in industrial production, and welding automation and robotization have become the trend of development. Improvement of welding quality is primary task of welding production automation. Therefore, the research on tracking control of welding process has practical significance.The tracking control problem of GTAW process has always been a hot topic in the field of welding research, and many global famous scholars have done a lot of research about it. In [1], some basic concepts relating to neural networks and how they can be used to model weld-bead geometry in terms of the equipment parameters selected to produce the weld are explained. The performance of neural networks for modeling is evaluated using actual welding data. It is concluded that the accuracy of neural network modeling is fully comparable with the accuracy achieved by more traditional modeling schemes. In arc welding processes, because of the complexity and nonlinearity of heat transfer phenomena, it is often difficult to design an effective control system based upon an exact mathematical model. In [2], a control system is designed to comprise a combined structure of the fuzzy SOC with conventional PD control for improving the system’s stability. Based on an analysis of accepted adaptive algorithms, an adaptive generalized predictive decoupling control scheme was constructed for GTAW process in [3]. In [4], a neurofuzzy model is used to model the welding process. Based on the dynamic fuzzy model, a predictive control system has been developed to control the welding process. Experiments confirmed that the developed control system is effective in achieving the desired fusion state despite the different disturbances. In [5], a PID controller was designed for the GTAW dynamic system model, and the Smith predictor was adopted to compensate time-delay at starting-welding. In order to achieve high quality control of the back weld width and the topside height of weld pool at the same time, a multi-input and multi-output adaptive fuzzy controller was designed by adding the fuzzy control to the adaptive controller to ensure the welding quality and precision [6]. In [7], a new type of fuzzy controller was designed by combining fuzzy control, neural network, and genetic algorithm in the control of GTAW process. Compared with traditional fuzzy control, the new controller had achieved better control effects. In [8], a nonlinear composite controller was designed for GTAW process. The peak current of the welding was adjusted according to the Hammerstein model, and the wire feeding speed was adjusted based on the fuzzy logic, which ensures that the back weld width and the topside height of weld pool were formed stably at the same time during the welding process. Since the precise mathematical model of welding process cannot be established, it is difficult to obtain satisfactory results by using classical and modern control theory. To overcome this problem, some researchers adopt model-free adaptive control (MFAC) to study the dynamic process of GTAW in [9]. Model-free adaptive controllers for single-input single-output, multi-input single-input, and multi-input multi-output model of welding process were designed, respectively, and good control results were achieved. In [10], the adaptive sliding mode control algorithm was applied to the automatic welding system. By changing the length of welding arc to control the heat that transferred to the welding workpiece, the control requirements were well met. In order to imitate the estimation and decision-making process of skilled welders, an adaptive neurofuzzy inference system (ANFIS) was used to establish the model that related the back weld width to the welding current and speed, and a model predictive control was used to follow a trajectory to reach the desired back weld width [11]. Although some achievements have been made in the above research, for welding process, which has multivariable, nonlinear, time-varying parameters and contains many uncertain factors and constraints, it is almost impossible to obtain an accurate mathematical model. Therefore, it is still difficult to achieve complete tracking control of welding process.In practical production, the process of automatic welding is a repetitive task. The repeatability is mainly reflected in the same welding workpiece and the same welding trajectory. However, the existing control methods have not considered how to use the repetitive information of welding process to improve the tracking performance of the system output. For repetitive systems, iterative learning control (ILC) [12–15] is an effective control method. The algorithm does not rely on the accurate mathematical model of the dynamic system, and it can achieve expected output trajectory tracking in limited time and interval with little prior knowledge and computational effort. The working principle of ILC is to use the previous iteration of the input information and the output error to modify control signal and gradually make the actual output converge to the desired output. In recent years, iterative learning control has not only made great progress in theory, but has also been widely used in many industrial fields [16–18], such as robotics, power, and industrial process control.For controlled objects whose system parameters do not change or change slowly, ILC can achieve good control effects. However, in practical welding production process, due to the extremely variable welding environment and the presence of noise interference and load disturbance, it is easy to cause the jumping change in parameters and even the structure of the welding system. In this case, ILC may not perform well and lead to the number of iterations increase, and more waste products are produced. According to the ILC theory, for the unknown dynamic systems, as long as convergence condition is satisfied, uniform tracking performance can be guaranteed even with partially known or estimated model. Though the exact model is not required, number of iteration for ILC with estimated model will increase compared with ILC with exact model. Multiple model adaptive control (MMAC) [19–21] is an effective method for the system with jumping parameters. According to the MMAC theory, multiple models will be built based on the uncertainties of controlled model’s parameters and structure. Multiple controllers will be designed according to multiple element models. According to the given switching mechanism, the model with most appropriate parameters to describe the plant will be decided, and the current controller will be switched to the controller corresponding to the best model [22, 23]. Therefore, in order to better meet the requirements of actual production, we introduce the idea of multimodel adaptive control into iterative learning control and design multiple model iterative learning control (MMILC) [24, 25]. Before every iteration, the best model is decided by the index switching function and corresponding controller based on ILC learning law is activated; thus the number of iterations will be reduced and the quality of the weld will be improved greatly.This paper is organized as follows. In Section2, we formulate the problem of GTAW process and welding model. In Section 3, we design iterative learning controller and multiple model iterative learning controller and analyze the convergence of the algorithm. Simulation and analysis are performed based on the welding model by using ILC and MMILC in Section 4. Finally, conclusions are given in Section 5.


## 2. Analysis of GTAW Process Control Problem

### 2.1. GTAW Process

GTAW is a kind of welding method in which tungsten rod is used as one electrode, base metal is used as the other electrode, and inert gas is used to protect the arc between the two electrodes. The schematic diagram of GTAW process is shown in Figure1. The tungsten electrode is clamped on the electrode holder and protrudes from the welding gun nozzle. The welding workpiece is welded by an arc that generated between the end of the tungsten electrode and the welding workpiece. After the beginning of the welding process, the arc conducts heat to the welded workpiece, which causes the temperature of the weld workpiece to rise rapidly, and then the weld workpiece is partially melted to form a weld pool. During the welding process, the nozzle emits protective gas around the tungsten electrode to avoid pollution to the welding process from the atmosphere. When it is necessary to fill the weld pool with metal, the welding wire can be fed into the weld pool at a certain speed from the front of the arc.Figure 1
Schematic diagram of GTAW.


### 2.2. System Model of GTAW

In the welding process, fusion penetration is the precondition to ensure the quality of welding. While back weld width, that is, the back width of weld pool, is the most direct expression of fusion penetration, so we choose the back weld width as the output of welding system. Many factors can influence the thermodynamic process to a great extent in welding, such as welding speed, welding current, wire feed rate, arc length, etc. These factors affect weld penetration, the size of the front and back pool, weld reinforcement, etc. Therefore, in theory, the above parameters can be quantified as input variables of the welding system. However, considering the controllability of parameters and the actual situation of the control system, the welding current and the wire feed rate are selected as the system inputs.The input-output dynamic characteristic model of GTAW system [26] is as follows:(1)yt=a1yt-1+a2yt-2+a3yt-3+a4yt-4+a5yt-5+b11u1t-1+b12u1t-2+b14u1t-4+b21u2t-1+b23u2t-3+b25u2t-5,where u1t,u2t are input variables, and they denote welding current and wire feed rate, yt is output variable, and it denotes the back weld width. This is a multi-input single-output system.


## 2.1. GTAW Process

GTAW is a kind of welding method in which tungsten rod is used as one electrode, base metal is used as the other electrode, and inert gas is used to protect the arc between the two electrodes. The schematic diagram of GTAW process is shown in Figure1. The tungsten electrode is clamped on the electrode holder and protrudes from the welding gun nozzle. The welding workpiece is welded by an arc that generated between the end of the tungsten electrode and the welding workpiece. After the beginning of the welding process, the arc conducts heat to the welded workpiece, which causes the temperature of the weld workpiece to rise rapidly, and then the weld workpiece is partially melted to form a weld pool. During the welding process, the nozzle emits protective gas around the tungsten electrode to avoid pollution to the welding process from the atmosphere. When it is necessary to fill the weld pool with metal, the welding wire can be fed into the weld pool at a certain speed from the front of the arc.Figure 1
Schematic diagram of GTAW.


## 2.2. System Model of GTAW

In the welding process, fusion penetration is the precondition to ensure the quality of welding. While back weld width, that is, the back width of weld pool, is the most direct expression of fusion penetration, so we choose the back weld width as the output of welding system. Many factors can influence the thermodynamic process to a great extent in welding, such as welding speed, welding current, wire feed rate, arc length, etc. These factors affect weld penetration, the size of the front and back pool, weld reinforcement, etc. Therefore, in theory, the above parameters can be quantified as input variables of the welding system. However, considering the controllability of parameters and the actual situation of the control system, the welding current and the wire feed rate are selected as the system inputs.The input-output dynamic characteristic model of GTAW system [26] is as follows:(1)yt=a1yt-1+a2yt-2+a3yt-3+a4yt-4+a5yt-5+b11u1t-1+b12u1t-2+b14u1t-4+b21u2t-1+b23u2t-3+b25u2t-5,where u1t,u2t are input variables, and they denote welding current and wire feed rate, yt is output variable, and it denotes the back weld width. This is a multi-input single-output system.


## 3. Establishment of Controller and Convergence Analysis

### 3.1. Single Model Iterative Learning Controller

For the dynamic characteristic model of GTAW system, a control input sequenceu1t,u2t,t∈0,T-1 is given. Assuming u1,2t=0(t<0), y(t)=0(t≤0), the response of system (1) is as follows:(2)y1=b11u10+b21u20=a0u0,y2=a1y1+b11u11+b12u10+b21u21=a1b11u10+b21u20+b11u11+b12u10+b21u21=b11u11+b21u21+a1b11+b12u10+a1b21u20=a0u1+a1u0.By analogy, we can get the following results:(3)y3=a0u2+a1u1+a2u0,⋮yT=a0uT-1+⋯+aT-1u0,where a(t) is a matrix consisting of one row and two columns, a0=[b11b21], a(1)=[a1b11+b12a1b21], u(t)=[u1(t)u2(t)]T.We defineU=u0,u1,…,u(T-1)T, y=y(1),y(2),⋯,y(T)T, then model (1) can be written in matrix form as follows:(4)y=LaU,where(5)La=a0a1a0⋮⋮aT-1aT-2⋯a0.Assuming that the impulse response sequence a of the system cannot be accurately known, ILC can be used to implement the tracking tasks.k denotes the number of iterations; we have(6)Uk=uk0,uk1,…,ukT-1T,yk=yk1,yk2,…,ykTT.At the kth iteration, model (4) can be written as(7)yk=LaUk.The following learning laws are used to obtain the control sequence:(8)uk+1t=ukt+∑s=0T-1csekt-s,where c(t) is gain coefficient, t∈[0,T-1].From (8), we have(9)uk+10uk+11⋮uk+1T-1=uk0uk1⋮ukT-1+c0c1c0⋮⋮cT-1cT-2⋯c0ek0ek1⋮ekT-1,where(10)Lc=c0c1c0⋮⋮cT-1cT-2⋯c0.Then the ILC law (8) can be written as follows:(11)uk+1=uk+Lcek,where ek=yd-yk, yd is the desired output of the system, yk is the actual output of the kth iteration, and L(c) is the learning gain matrix which will be chosen.


### 3.2. Multiple Model Iterative Learning Controller

The dynamic process of welding is a complex object with multiple variable, multiple linear, time-varying parameters, and many uncertain factors. For the time-varying characteristics of welding system parameters, we consider the jump of system parameters which is a special case of time-varying parameters. Using the idea of multimodel, we establish multiple models to cover all jump parameters and design multimodel iterative learning controller to improve the control accuracy of welding system in the case of parameters jump.(  1) Establishment of Multiple Welding Models and Their Corresponding Controllers (12) M = M i ∣ i = 1,2 , ⋯ , n ,where Mi denotes n welding models, and it can be written as(13)yt=a1iyt-1+a2iyt-2+a3iyt-3+a4iyt-4+a5iyt-5+b11iu1t-1+b12iu1t-2+b14iu1t-4+b21iu2t-1+b23iu2t-3+b25iu2t-5.The corresponding controllers are denoted as follows:(14)C=Cj∣j=1,2,⋯,m,where Cj denotes controllers corresponding to the model Mi, and the controller of each model can make the corresponding model achieve good control performance. Besides, each controller can guarantee the stability of the system. Cj can be written as(15)uk+1=uk+Lcek,where ek=yd-yk.(  2) Establishment of Index Switching Function (16) L i , k = ∑ m = 1 T β T - m e k i m ,  0 < β ≤ 1 ,where eki(t)=yk(t)-yki(t), yk(t) denotes the ideal output of the system, yki(t) denotes the output of the ith model, and β denotes weighting factor.Remarks. By establishing multiple models, the transient response of the system can be improved and the number of iterations can be reduced.The switching steps of MMILC are as follows:(1) Multiple models, corresponding controllers, and index switching functions are established.(2) At the end of thekth iteration, the best model to describe the current environment will be selected based on index switching functions; i.e., model Ml(k) is the best model, l(k) is defined by(17)lk=arg⁡min1≤i≤n⁡∑m=1TβT-mekim.(3) In the next iteration, it will switch to controller Cl(k) based on model Ml(k).


### 3.3. Convergence Analysis of System with Fixed Parameters

Lemma 1 (see [20]).
ForA∈Cn×n with its eigenvalues λi,i∈1,2,⋯,n, if maxi∈1,2,⋯,n⁡λi<δ<1, the following inequality can be guaranteed:(18)Ak2<σγk,where σ is a positive constant and δ<γ<1.Proof.
For an arbitrary positiveε, there is a compatible norm·∗which satisfies(19)A∗<max⁡λi+ε<δ+ε.Due to δ<1, we can let the ε satisfy δ+ε<1, and let γ=δ+ε, then(20)Ak∗<A∗⋯A∗︸k<γk<1.From the functional analysis, compatible norm satisfies(21)Ak2<σAk∗,where σ is a positive number which is determined by the relationship between 2-norm and ∗-norm.
Hence(22)Ak2<σγk.This completes the proof.Theorem 2.
The errorek converges to zero exponentially for the arbitrary initial error e0 as k→∞ if the coefficient c(0) satisfies(23)1-a0c0<1.Proof.
From model (7) and ILC law (11), we have(24)ek+1=yd-Lauk+1=yd-Lauk-LaLcek=I-LaLcek,where La is a matrix with T rows and 2T columns and Lc is a matrix with 2T rows and T columns.
From the equation ofLa and Lc, we can easily show that the matrix LaLc is a lower triangular Toeplitz matrix with T rows and T columns. So I-LaLc is also a lower triangular Toeplitz matrix, and its first column is given as follows:(25)fc∣a=IT-Lac,where c=c0,c1,…,c(T-1)T and IT=[10⋯0]T is a T dimensional vector, so we can easily know that all eigenvalues of the lower triangular Toeplitz matrix I-LaLc are given by 1-a(0)c(0).
TakingA=I-LaLc, (24) can be written as(26)ek=Aek-1=Ake0.Taking norms for both sides of (26) to get(27)ek=Ak∙e0.Taking λ=1-a(0)c(0), according to Lemma 1, if the eigenvalues of A satisfy maxi∈1,2,⋯,n⁡λi<δ<1, Ak2<σγk can be guaranteed. So (27) can be written as(28)ek<σγke0,where σ is a positive constant, δ<γ<1. So when k→∞, we can obtain ek→0.
This completes the proof.


### 3.4. Convergence Analysis of System with Jumping Parameters

Theorem 3.
If the system parameters jump during the iterative learning process, the errorek converges to zero exponentially for the arbitrary initial error e0 as k→∞ if the following conditions can be satisfied(1)1-a(0)cj(0)<1,j=1,2,…,m.(2) System parameters jump among finite values.Proof.
The proof can be performed similarly as in the proof of Theorem2.
Assuming that the system model parameters jump amongm values in the iterative learning process, (26) can be written as follows:(29)ek=A0km∙A1km-1∙⋯∙Am-1k1∙Amk0e0,where k0+k1+⋯+km-1+km=k,k is iteration number.
Take norms for both sides of (29) to get(30)ek=A0km∙A1km-1∙⋯∙Am-1k1∙Amk0e0.From Lemma 1, if the eigenvalues of A0,A1,…,Am satisfy maxi∈1,2,⋯,n⁡λi<δ<1, we have(31)A0km∙A1km-1∙⋯∙Am-1k1∙Amk0<σ0γkm∙σ1γkm-1∙⋯∙σm-1γk1∙σmγk0=σ0σ1⋯σm-1σm∙γkmγkm-1⋯γk1γk0.Substitute (31) into (30); we obtain(32)ek<σ0σ1⋯σm-1σm∙γkmγkm-1⋯γk1γk0e0.According to Lemma 1, we can easily know that σ0σ1⋯σm-1σm is a positive constant. If m is a finite value, then σ0σ1⋯σm-1σm is a bounded positive constant. So when k→∞, we can obtain γkmγkm-1⋯γk1γk0→0, then ek→0.
This completes the proof.


## 3.1. Single Model Iterative Learning Controller

For the dynamic characteristic model of GTAW system, a control input sequenceu1t,u2t,t∈0,T-1 is given. Assuming u1,2t=0(t<0), y(t)=0(t≤0), the response of system (1) is as follows:(2)y1=b11u10+b21u20=a0u0,y2=a1y1+b11u11+b12u10+b21u21=a1b11u10+b21u20+b11u11+b12u10+b21u21=b11u11+b21u21+a1b11+b12u10+a1b21u20=a0u1+a1u0.By analogy, we can get the following results:(3)y3=a0u2+a1u1+a2u0,⋮yT=a0uT-1+⋯+aT-1u0,where a(t) is a matrix consisting of one row and two columns, a0=[b11b21], a(1)=[a1b11+b12a1b21], u(t)=[u1(t)u2(t)]T.We defineU=u0,u1,…,u(T-1)T, y=y(1),y(2),⋯,y(T)T, then model (1) can be written in matrix form as follows:(4)y=LaU,where(5)La=a0a1a0⋮⋮aT-1aT-2⋯a0.Assuming that the impulse response sequence a of the system cannot be accurately known, ILC can be used to implement the tracking tasks.k denotes the number of iterations; we have(6)Uk=uk0,uk1,…,ukT-1T,yk=yk1,yk2,…,ykTT.At the kth iteration, model (4) can be written as(7)yk=LaUk.The following learning laws are used to obtain the control sequence:(8)uk+1t=ukt+∑s=0T-1csekt-s,where c(t) is gain coefficient, t∈[0,T-1].From (8), we have(9)uk+10uk+11⋮uk+1T-1=uk0uk1⋮ukT-1+c0c1c0⋮⋮cT-1cT-2⋯c0ek0ek1⋮ekT-1,where(10)Lc=c0c1c0⋮⋮cT-1cT-2⋯c0.Then the ILC law (8) can be written as follows:(11)uk+1=uk+Lcek,where ek=yd-yk, yd is the desired output of the system, yk is the actual output of the kth iteration, and L(c) is the learning gain matrix which will be chosen.


## 3.2. Multiple Model Iterative Learning Controller

The dynamic process of welding is a complex object with multiple variable, multiple linear, time-varying parameters, and many uncertain factors. For the time-varying characteristics of welding system parameters, we consider the jump of system parameters which is a special case of time-varying parameters. Using the idea of multimodel, we establish multiple models to cover all jump parameters and design multimodel iterative learning controller to improve the control accuracy of welding system in the case of parameters jump.(  1) Establishment of Multiple Welding Models and Their Corresponding Controllers (12) M = M i ∣ i = 1,2 , ⋯ , n ,where Mi denotes n welding models, and it can be written as(13)yt=a1iyt-1+a2iyt-2+a3iyt-3+a4iyt-4+a5iyt-5+b11iu1t-1+b12iu1t-2+b14iu1t-4+b21iu2t-1+b23iu2t-3+b25iu2t-5.The corresponding controllers are denoted as follows:(14)C=Cj∣j=1,2,⋯,m,where Cj denotes controllers corresponding to the model Mi, and the controller of each model can make the corresponding model achieve good control performance. Besides, each controller can guarantee the stability of the system. Cj can be written as(15)uk+1=uk+Lcek,where ek=yd-yk.(  2) Establishment of Index Switching Function (16) L i , k = ∑ m = 1 T β T - m e k i m ,  0 < β ≤ 1 ,where eki(t)=yk(t)-yki(t), yk(t) denotes the ideal output of the system, yki(t) denotes the output of the ith model, and β denotes weighting factor.Remarks. By establishing multiple models, the transient response of the system can be improved and the number of iterations can be reduced.The switching steps of MMILC are as follows:(1) Multiple models, corresponding controllers, and index switching functions are established.(2) At the end of thekth iteration, the best model to describe the current environment will be selected based on index switching functions; i.e., model Ml(k) is the best model, l(k) is defined by(17)lk=arg⁡min1≤i≤n⁡∑m=1TβT-mekim.(3) In the next iteration, it will switch to controller Cl(k) based on model Ml(k).


## 3.3. Convergence Analysis of System with Fixed Parameters

Lemma 1 (see [20]).
ForA∈Cn×n with its eigenvalues λi,i∈1,2,⋯,n, if maxi∈1,2,⋯,n⁡λi<δ<1, the following inequality can be guaranteed:(18)Ak2<σγk,where σ is a positive constant and δ<γ<1.Proof.
For an arbitrary positiveε, there is a compatible norm·∗which satisfies(19)A∗<max⁡λi+ε<δ+ε.Due to δ<1, we can let the ε satisfy δ+ε<1, and let γ=δ+ε, then(20)Ak∗<A∗⋯A∗︸k<γk<1.From the functional analysis, compatible norm satisfies(21)Ak2<σAk∗,where σ is a positive number which is determined by the relationship between 2-norm and ∗-norm.
Hence(22)Ak2<σγk.This completes the proof.Theorem 2.
The errorek converges to zero exponentially for the arbitrary initial error e0 as k→∞ if the coefficient c(0) satisfies(23)1-a0c0<1.Proof.
From model (7) and ILC law (11), we have(24)ek+1=yd-Lauk+1=yd-Lauk-LaLcek=I-LaLcek,where La is a matrix with T rows and 2T columns and Lc is a matrix with 2T rows and T columns.
From the equation ofLa and Lc, we can easily show that the matrix LaLc is a lower triangular Toeplitz matrix with T rows and T columns. So I-LaLc is also a lower triangular Toeplitz matrix, and its first column is given as follows:(25)fc∣a=IT-Lac,where c=c0,c1,…,c(T-1)T and IT=[10⋯0]T is a T dimensional vector, so we can easily know that all eigenvalues of the lower triangular Toeplitz matrix I-LaLc are given by 1-a(0)c(0).
TakingA=I-LaLc, (24) can be written as(26)ek=Aek-1=Ake0.Taking norms for both sides of (26) to get(27)ek=Ak∙e0.Taking λ=1-a(0)c(0), according to Lemma 1, if the eigenvalues of A satisfy maxi∈1,2,⋯,n⁡λi<δ<1, Ak2<σγk can be guaranteed. So (27) can be written as(28)ek<σγke0,where σ is a positive constant, δ<γ<1. So when k→∞, we can obtain ek→0.
This completes the proof.


## 3.4. Convergence Analysis of System with Jumping Parameters

Theorem 3.
If the system parameters jump during the iterative learning process, the errorek converges to zero exponentially for the arbitrary initial error e0 as k→∞ if the following conditions can be satisfied(1)1-a(0)cj(0)<1,j=1,2,…,m.(2) System parameters jump among finite values.Proof.
The proof can be performed similarly as in the proof of Theorem2.
Assuming that the system model parameters jump amongm values in the iterative learning process, (26) can be written as follows:(29)ek=A0km∙A1km-1∙⋯∙Am-1k1∙Amk0e0,where k0+k1+⋯+km-1+km=k,k is iteration number.
Take norms for both sides of (29) to get(30)ek=A0km∙A1km-1∙⋯∙Am-1k1∙Amk0e0.From Lemma 1, if the eigenvalues of A0,A1,…,Am satisfy maxi∈1,2,⋯,n⁡λi<δ<1, we have(31)A0km∙A1km-1∙⋯∙Am-1k1∙Amk0<σ0γkm∙σ1γkm-1∙⋯∙σm-1γk1∙σmγk0=σ0σ1⋯σm-1σm∙γkmγkm-1⋯γk1γk0.Substitute (31) into (30); we obtain(32)ek<σ0σ1⋯σm-1σm∙γkmγkm-1⋯γk1γk0e0.According to Lemma 1, we can easily know that σ0σ1⋯σm-1σm is a positive constant. If m is a finite value, then σ0σ1⋯σm-1σm is a bounded positive constant. So when k→∞, we can obtain γkmγkm-1⋯γk1γk0→0, then ek→0.
This completes the proof.


## 4. Simulation Research and Analysis

The mathematical model and model parameters of the welding system in this paper are derived from an actual welding system [9, 26, 27]. The structure of GTAW system is shown in Figure 2. The system mainly includes central control computer, welding power source, wire feeder, wire feeding control box, welding gun, gas flow controller, etc. Under the control of the central controller, the entire system can perform functions such as arc starting, welding, arc stopping, and walking. During the welding process, the back weld width is controlled by adjusting welding current and wire feed rate to ensure the welding quality. The welding current and the wire feed rate are controlled by adjusting the input voltage of the control end of welding power source. The peripheral equipment includes traveller and conveying mechanism. The conveying mechanism has two mutually perpendicular motion shafts, which can complete various curve movements in the plane.Figure 2
Structure diagram of GTAW system.The mathematical model of GTAW system is as follows:(33)yt=a1yt-1+a2yt-2+a3yt-3+a4yt-4+a5yt-5+b11u1t-1+b12u1t-2+b14u1t-4+b21u2t-1+b23u2t-3+b25u2t-5,The system parameters are as follows:(34)a1a2a3a4a5=1.2245,-0.7935,0.45269,-0.23124,0.11518,b11b12b14=0.0085696,-0.3748,0.0039714,b21b23b25=-0.16826,0.0023674,-0.079501.In order to show the effectiveness of our results, two cases are considered in this section.


### 4.1. ILC for Welding System with Constant Parameters

Takingc(1)c(2)⋯c(T-1)=00⋯0, we have(35)LC=c00c0⋮⋮⋱00⋯c0.Substitute (35) into (11), the ILC law (11) can be written as(36)uk+1t=ukt+c0ekt.The matrix a(0)=b11b21 = 0.0085696 −0.16826 is given. Set the learning gain as c(0)=-0.05 −0.2T, then it follows that 1-a(0)c(0)=0.96<1. The desired value of back weld width is given as yd(k) = 6 mm.Figure3 gives the tracking effect of back weld width at 5th, 10th, and 30th iterations, respectively. It shows that the actual output of the system converges to the desired output quickly within 30 iterations. Moreover, the performance of the maximal tracking error is presented in Figure 4. It shows that the maximal tracking error converges to zero gradually with the increase of the number of iterations, which means that the back weld width will fully meet the desired value. It proves the effectiveness of ILC for GTAW process control.Figure 3
Tracking curve of back weld width.Figure 4
Maximal tracking error in iterative domain.


### 4.2. Welding System with Jumping Parameters

The above simulation results in Section4.1 are obtained under the condition that the welding system parameters are fixed. In this section, for the situation that the system parameters jump during the welding process, simulation and analysis are carried out. We assume that the real parameter b21 of the welding model jumps during the welding process, and the jumping parameters are as follows:(37)b21=-0.57k<5-0.375≤k≤10-0.16k>10t∈1,T.In modeling process, the model parameters we get are often inaccurate. We randomly select several estimation ofb21 around the true value for simulation. Three estimated values of the parameters are given as b211=-0.67, b212=-0.42, and b213=-0.17, corresponding to model 1, model 2, and model 3. For system with jumping parameter, we consider two cases by using single model ILC and multiple model ILC.Case 1 (single model ILC).
For the system with jumping parameters, the simulation results using single model ILC are shown in Figures5 and 6. Figure 5 gives the tracking effect of back weld width at 8th, 15th, 60th iterations, respectively. It shows that the tracking effect of the 15th iteration is worse than that of the 8th iteration because of the jump of system parameters, and after 60 iterations, the actual output converges to the desired output gradually. The performance of the maximal tracking error in each welding process is presented in Figure 6. From Figure 6, we can see that the tracking error fluctuates greatly and the tracking effect is poor due to the jump of system parameters in the 5th and 10th iterations.Figure 5
Tracking curve of back weld width.Figure 6
Maximal tracking error in iterative domain.Case 2 (multiple model ILC).
When multiple model ILC is applied to the welding system with jumping parameters, the simulation results which are shown in Figures7, 8, and 9, and Figure 9 depict the switching procedure that how the best model is selected at every iteration when the parameter jumps, i.e., when the parameter b21 changes from -0.57 to -0.37 at the 5th iteration, model 2 is calculated to be the precise one instead of model 1, so model 2 is selected when the 6th iteration begins. Through this control strategy, after about 25 iterations, the tracking error converges to zero gradually, and the performance of the maximal tracking error is shown in Figures 7 and 8. Compared with single model ILC, MMILC can decrease the number of iterations effectively and improve the tracking effect.Figure 7
Tracking curve of back weld width.Figure 8
Maximal tracking error in iterative domain.Figure 9
Switching procedure of model.


## 4.1. ILC for Welding System with Constant Parameters

Takingc(1)c(2)⋯c(T-1)=00⋯0, we have(35)LC=c00c0⋮⋮⋱00⋯c0.Substitute (35) into (11), the ILC law (11) can be written as(36)uk+1t=ukt+c0ekt.The matrix a(0)=b11b21 = 0.0085696 −0.16826 is given. Set the learning gain as c(0)=-0.05 −0.2T, then it follows that 1-a(0)c(0)=0.96<1. The desired value of back weld width is given as yd(k) = 6 mm.Figure3 gives the tracking effect of back weld width at 5th, 10th, and 30th iterations, respectively. It shows that the actual output of the system converges to the desired output quickly within 30 iterations. Moreover, the performance of the maximal tracking error is presented in Figure 4. It shows that the maximal tracking error converges to zero gradually with the increase of the number of iterations, which means that the back weld width will fully meet the desired value. It proves the effectiveness of ILC for GTAW process control.Figure 3
Tracking curve of back weld width.Figure 4
Maximal tracking error in iterative domain.


## 4.2. Welding System with Jumping Parameters

The above simulation results in Section4.1 are obtained under the condition that the welding system parameters are fixed. In this section, for the situation that the system parameters jump during the welding process, simulation and analysis are carried out. We assume that the real parameter b21 of the welding model jumps during the welding process, and the jumping parameters are as follows:(37)b21=-0.57k<5-0.375≤k≤10-0.16k>10t∈1,T.In modeling process, the model parameters we get are often inaccurate. We randomly select several estimation ofb21 around the true value for simulation. Three estimated values of the parameters are given as b211=-0.67, b212=-0.42, and b213=-0.17, corresponding to model 1, model 2, and model 3. For system with jumping parameter, we consider two cases by using single model ILC and multiple model ILC.Case 1 (single model ILC).
For the system with jumping parameters, the simulation results using single model ILC are shown in Figures5 and 6. Figure 5 gives the tracking effect of back weld width at 8th, 15th, 60th iterations, respectively. It shows that the tracking effect of the 15th iteration is worse than that of the 8th iteration because of the jump of system parameters, and after 60 iterations, the actual output converges to the desired output gradually. The performance of the maximal tracking error in each welding process is presented in Figure 6. From Figure 6, we can see that the tracking error fluctuates greatly and the tracking effect is poor due to the jump of system parameters in the 5th and 10th iterations.Figure 5
Tracking curve of back weld width.Figure 6
Maximal tracking error in iterative domain.Case 2 (multiple model ILC).
When multiple model ILC is applied to the welding system with jumping parameters, the simulation results which are shown in Figures7, 8, and 9, and Figure 9 depict the switching procedure that how the best model is selected at every iteration when the parameter jumps, i.e., when the parameter b21 changes from -0.57 to -0.37 at the 5th iteration, model 2 is calculated to be the precise one instead of model 1, so model 2 is selected when the 6th iteration begins. Through this control strategy, after about 25 iterations, the tracking error converges to zero gradually, and the performance of the maximal tracking error is shown in Figures 7 and 8. Compared with single model ILC, MMILC can decrease the number of iterations effectively and improve the tracking effect.Figure 7
Tracking curve of back weld width.Figure 8
Maximal tracking error in iterative domain.Figure 9
Switching procedure of model.


## 5. Conclusions

Aiming at the repeatability of automatic welding, the welding process is studied based on ILC algorithm in this paper. Considering that the welding process is a complex nonlinear process with multiple variable, multiple linear, and time-varying parameters and the system parameters jump easily due to external disturbances in the welding process, we combine MMAC with ILC. Firstly, the GTAW process and model are introduced. Then according to the model of GTAW process, an iterative learning controller and a multiple model iterative learning controller are designed. And the convergence of the algorithm is proved. The simulation results show the effectiveness of the algorithm.



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*Source: 6137352-2019-05-16.xml*