Nonlinear model identification of dissimilar laser joining of S.S 304 and ABS using the Hammerstein

Optik - International Journal for Light and Electron Optics 225 (2021) 165649

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Original research article

Nonlinear model identification of dissimilar laser joining of S.S 304 and ABS using the Hammerstein–Wiener method Quyen Nguyen a, Seyed Amin Bagherzadeh b, Amir Parsian b, Mohammad Akbari b, Arash Karimipour b, *, Amirhosein Mosavi c, d, ** a

Institute of Research and Development, Duy Tan University, Danang 550000, Viet Nam Department of Mechanical Engineering, Najafabad Branch, Islamic Azad University, Najafabad, Iran c Environmental Quality, Atmospheric Science and Climate Change Research Group, Ton Duc Thang University, Ho Chi Minh City, Viet Nam d Faculty of Environment and Labour Safety, Ton Duc Thang University, Ho Chi Minh City, Viet Nam b

A R T I C L E I N F O

A B S T R A C T

Keywords: Hammerstein–Wiener method Pulsed laser welding Root-mean-square error Experimental results Training and test

The nonlinear system identification Hammerstein–Wiener method is used according to the experimental outputs of laser joining of Acrylonitrile Butadiene Styrene (ABS) and stainless steel 304 concerning the temperature distribution in different parameters of a pulsed laser, including welding speed, focal length, pulse duration, frequency, power and current. Limited linear and nonlinear models exist for such processes; however, due to restrictions and complex numerical calculations, their use in the real-time process control is not practical. An accurate model for the dissimilar laser joining of metals and plastic materials is essential for controller design and process automation. Compared to other popular nonlinear compensation techniques, this model has a simpler structure and needs fewer calculations. This paper considers the identification problems of Hammerstein–Wiener nonlinear systems by employing the gradient search technique.

Seven available data sets for training and test phases in dissimilar input parameters were used, and root-mean-square error (RMSE), mean absolute percentage error (MAPE) and fitness value were calculated. The results indicated that the identified Hammerstein-Wiener model had acceptable performance in training and test phases and was in good agreement with experimental results.

1. Introduction The laser welding is a growing technology, which in recent years, has received considerable attention. This technology is well known for its low distortion, high ratio of depth to width, high speed of welding and its deep penetration [1–8]. Nevertheless, environmental concerns and ineffective quality monitoring limited the development of automated laser welding in industrial usage.

One of the most important dilemmas in laser welding technology is setting the parameters such as pulse duration, the focus point, and the used power rate [9]. Weld pool geometry also has a considerable effect on welding quality [10–13]; therefore, investigating the connection between the mentioned parameters and the weld pool geometry can be helpful to recognize the welding process aspects and

* Corresponding author. ** Corresponding author at: Environmental Quality, Atmospheric Science and Climate Change Research Group, Ton Duc Thang University, Ho Chi Minh City, Viet Nam. E-mail addresses: arashkarimipour1981@gmail.com (A. Karimipour), amirhosein.mosavi@tdtu.edu.vn (A. Mosavi). https://doi.org/10.1016/j.ijleo.2020.165649 Received 17 June 2020; Received in revised form 20 August 2020; Accepted 17 September 2020 Available online 22 September 2020 0030-4026/© 2020 Published by Elsevier GmbH.

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Fig. 1. Wiener–Hammerstein (sandwich) system.

Fig. 2. Schematic diagram of the welding configuration [14].

find a suitable model to control this parameter. Numerous researches have been conducted to monitor the weld pool condition. Some focused on heat distribution [14,15], some concentrated on emission signals [7,16,17], and others targeted the weld pool images attained with charge-coupled device cameras [12,18]. It can be stated that laser welding is a complicated physicochemical and thermodynamic process, including evaporating, plasma, material melting, heat affected zone keyhole occurrence, forming and so forth. Usually, some linear or nonlinear analytical and numerical methods are used to simulate actual industrial applications [19–24].

Additionally, the models introduced for the laser welding process are linear models; however, some parameters can have a nonlinear effect on the welding process and influence the system performance [25]. Thus, to achieve better results it is better than laser welding be dealt with as a nonlinear process. It is a common practice to use a linear model to estimate a nonlinear system; however, the operation region in control system designed by the linear model has a limited range. Accordingly, a nonlinear model based on input-output data is preferred to comprehend the global behaviour of the laser welding process and consequently to obtain a higher operation region for the system. Block-oriented systems have been widely used in system modelling and engineering practice [26–29].

Wiener and the Hammerstein structure is one of the representatives of this class of objects, which may have the combination of linear dynamic and nonlinear static blocks. Both linear and non-linear models possess decent approximation properties and are considerably discussed in the literature [30–35]. In this study, we focus on the Wiener–Hammerstein (LNL) system that inherits the properties from both structures. The Wiener–Hammerstein model separates the linear and nonlinear parts of the system. In this model, the input and output nonlinear parts are static, while the middle linear part is dynamic. Therefore, Wiener–Hammerstein models are flexible enough to capture complicated nonlinear and dynamic phenomena in real processes. Furthermore, due to its block representation, it is straightforward to implement this method for various problems. Owing to these advantages, the Wiener–Hammerstein method has been widely used for black-box and grey-box system identification problems [36–38].

Fig. 1 shows a set of two linear dynamic blocks and internal static nonlinear block connections. The LNL system can be demanding in the identification stage, usually due to the correlation of nonlinearity input caused by the first dynamic subsystem. Therefore, for the LNL system, very limited identification algorithms have been analysed and proposed [39]. Most of the approaches proposed in the mentioned works employ different and specific approaches such as the aggregative modelling approach [17], Gaussian input (BLA) [35,40], Volterra and Wiener series expansions [25,41], gradient methods [42] and the maximum-likelihood based method [43].

Numerous studies on Wiener–Hammerstein models have been conducted to reduce prior restrictions imposed on system characteristics and input excitation [36,44–48]. Most nonlinear methods in the literature are based on discrete models; however, for advanced analysis, nonlinear systems are required. Hence for the first time, the Hammerstein-Wiener method was utilized to discrete a proper model for the dissimilar laser joining of S.S 304 and ABS. Thus, the main contributions of the paper are proposing an algorithm to identify a nonlinear subsystem almost independent of dynamic subsystems properties and using this algorithm for diverse input signals.

2. Statement of the problem The nonlinear equalization Hammerstein–Wiener method is evaluated according to the experimental results obtained by Peng et al.

[14] concerning the temperature distribution during the laser bonding of ABS and S.S 304 in different parameters of pulsed laser containing speed of laser welding (S.W = 4.2–8.2 mm/s), pulse duration (Dp = 6− 10 ms), focal length (F.L = 1− 5 mm), frequency (f = 10− 20 Hz), current (I = 100–150 A) and power (P = 178− 256 W). They selected S.S 304 and ABS sheets with specified di­ mensions and 1 mm and 2 mm in thickness, respectively. They used pulsed Nd:YAG laser model IQL-10 and measured the temperature by thermocouple type K with 1 mm in diameter installed in a 2-mm lateral distance from the centre of the melt pool (see Fig. 2).

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Fig. 3. The Hammerstein-Wiener structure.

3. Numerical method Normally, the nonlinear system identification methods fall into two groups, namely the parametric and non-parametric methods. If the questioned system’s structure is available as a previous assumption, the identification problem may be reduced to a parameter approximation problem, so that it is basically an optimization problem. Thus, any of approaches such as black, grey and white box tactics can be employed depending on whether a model description of general ready-made or a tailor-made model set by incorporating a priori structural info is employed. Alternatively, if limited a priori information is available on the structure of the system, the identification method is non-parametric. The non-parametric approaches usually need more sophisticated optimization techniques than the parametric ones demand. A method of non-parametric system identification is referred to as any technique aiming at pre­ dicting complex behaviours of a system without selecting a confined set of possible regressors. Therefore, it does not use explicitly a finite-dimensional parameter vector.

The dissimilar laser welding process is a complex phenomenon due to the following reasons: It is an inherently nonlinear system in which the connection amongst the inputs and outputs of the system are highly nonlinear.

It is a dynamic system, namely a system whose outputs are functions of time.

The system has memory; in other words, the system outputs are functions of previous values of the system inputs and outputs.

It is a multi-input system; in other words, several parameters affect the outputs. It is very difficult to separate the effects of an individual input on the system output.

• The system behaviour should be investigated over various range of input parameters values. Thus, it is impossible to find operating points and linearize the model around them.

• • • •

Hence, nonlinear model identification is required to determine the connection between the inputs and outputs of the system to predict the system behaviour at unknown inputs. There are several nonlinear identification methods. In this paper, the HammersteinWiener method is examined to identify the dissimilar laser joining of S.S 304 and ABS. The Hammerstein-Wiener is a mathematical model to identify nonlinear dynamic phenomena. The Hammerstein-Wiener aims to interpret the system as some parallel linear and nonlinear parts in which time-variant phenomena are captured by the linear parts and the nonlinearities are captured by the nonlinear parts. There are several nonlinearity estimators for the Hammerstein-Wiener method, such as piecewise linear function, wavelet network, one-layer sigmoid network, dead zone, saturation and one-dimensional polynomial. Furthermore, the transfer functions of the linear parts can have numerators and denominators with dissimilar orders. Therefore, the Hammerstein-Wiener method has the potential to be implemented to identify various nonlinear dynamic systems. The Hammerstein-Wiener method is usually employed as a black-box identification method. However, it may be used as a grey-box identification method when there is an insight into the physical phenomena governing the system. The model of Hammerstein-Wiener model comprises two input and output static nonlinear blocks on both sides of a dynamic linear block. Fig. 3 illustrates the structure of the Hammerstein-Wiener model. The linear block is typically a transfer function. Therefore, the model can be represented as follows: w(t) = f (u(t) ) x(t) =

B(q) w(t) F(q)

(1)

y(t) = h(x(t) ) where B(q) and F(q) are polynomials.

This model requires a training phase in which the model parameters are estimated so that the error between the estimated and measured outputs are minimized. To that end, an iterative process is needed. Until now, several methods have been used to acquire the parameters of the linear and nonlinear parts of the Wiener–Hammerstein models. In this paper, the default “nlhw” subroutine in MATLAB is used in which the model parameters are iteratively estimated to minimize the error between the predicted and real outputs.

To that end, the gradient search scheme is employed to find a local minimum in the search space. Suppose that F(x) should be minimized in a defined domain. If F(x) is differentiable around point an , the gradient search scheme tries to find the next immediate minimum point as follows: an+1 = an − γ∇F(an ) ,

(2)

n≥0

In which the step size γis a positive real number. Since this method is a greedy one, it may be trapped in local minima. Therefore, some most effective techniques can be employed in this regard such as the fractional gradient [49–51].

Once the model is trained, it can predict the outputs for any arbitrary combination of inputs after selecting suitable initial con­ ditions. Usually, the available data are divided into training and test datasets. The training dataset is employed at the training phase to 3

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Table 1 Available datasets for the training and test phases [14].

Test number

Dp (ms)

f (Hz)

I (A)

P (W)

#1 #2 #3 #4 #5 #6 #7

8 10 10 8 6 10 8

15 10 15 20 15 20 15

130 130 130 130 130 100 150

215 192 250 225 178 209 256

Fig. 4. The identified input nonlinearity estimators.

determine the model parameters, while the test dataset is utilized to evaluate the model performance after training. In other words, the test dataset is not used to model but to validate the obtained model.

4. Results and discussion In this paper, identification of a nonlinear model of the dissimilar laser joining of S.S 304 and ABS is performed using the Hammerstein-Wiener method. There are 7 available datasets for training and test phases at dissimilar pulse duration, frequency, current and power. Table 1 presents the list of available datasets. It can be seen that there are four inputs (i.e., the pulse duration, frequency, current and power) and one output (i.e., temperature). In all considered experiments, the speed of welding is 6.2 mm/s, and the focal length is 5 mm. The first 6 datasets are used at the training phase, while the last one is utilized for the test phase [14].

A Hammerstein-Wiener model is employed with 10-unit piecewise linear functions as input and output nonlinearity estimators. The linear part includes four transfer functions corresponding to inputs with two zeros, three poles and one input delay. Fig. 4 presents the identified input nonlinearity estimators. Fig. 5 illustrates the positions of the poles and zeros of the transfer functions (i.e., the linear block). Moreover, Fig. 6 shows the identified output nonlinearity estimator.

For verification, the identified Hammerstein-Wiener model is employed to check the predicted output in the training dataset. The results are illustrated in Fig. 7 which shows a good agreement between the predicted and measure outputs.

The following performance criteria are used to check the fidelity of the identified model:

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Fig. 5. The positions of the poles and zeros of the transfer functions.

Fig. 6. The identified output nonlinearity estimator.

(3) The RMSE is defined as follows: RMSE =

( )1/2 n 1∑ (yi − ̂y i )2 n i=1

(3)

(4) The MAPE is defined as follows: ⃒ n ⃒ ⃒y i − ̂ 100% ∑ y i ⃒​⃒ ⃒ MAPE = n i=1 ⃒ yi ⃒

(4)

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Fig. 7. Verification of the predicted outputs against the measured ones in the training dataset for the test (a) #1, (b) #2, (c) #3, (d) #4, (e) #5 and (f) #6.

Table 2 The values of the performance criteria in the training phase.

Training

Mean

Test number

RMSE

MAPE %

fitness %

1 2 3 4 5 6

3.8608 2.4745 5.7600 5.5069 6.9768 4.8452 4.9040

5.8909 3.9528 7.0881 6.1175 7.2294 5.7243 6.0005

97.8826 98.3509 97.4684 90.8555 83.0737 95.9428 93.9289

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Fig. 8. The predicted outputs against the measured ones in the test phase.

Table 3 The values of the performance criteria in the test phase.

Phase

Test number

RMSE

MAPE %

fitness %

Test

7

10.1773

14.3003

82.7396

(5) The fitness value is described as follows: ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ fitness = 100%⎜ ⎜1 − ⎜ ⎜ ⎜ ⎜ ⎝

⎞

⎟ ⎟ ⎟ ⎟ 2 ⎟ (yi − ̂y 1 ) ⎟ i=1 ⎟ ⎛ ⎛ ⎞2 ⎞1/2 ⎟ ⎟ n ∑ ⎟ yi n ⎜ ⎟ ⎜∑ ⎟ ⎟ ⎜ ⎜yi − i=1 ⎟ ⎟ ⎟ ⎠ n ⎠ ⎠ ⎝ i=1 ⎝ (

n ∑

)1/2

(5)

where y and ̂ y are the measured and estimated outputs, respectively. Table 2 presents the list of the values of the performance criteria for verification cases.

When the model is trained, it can predict the outputs for any arbitrary combination of inputs after selecting suitable initial con­ ditions. For the test phase, the identified Hammerstein-Wiener model is employed to predict the output in the non-training dataset #7.

Fig. 8 and Table 3 present the results.

Based on the results, one can observe that the identified Hammerstein-Wiener model has acceptable performance in training and test phases. In other words, the precision and generalization of the model are acceptable. Hence, the Hammerstein-Wiener method is successful in identifying the nonlinear and time-variant phenomenon of dissimilar laser joining of S.S 304 and ABS examined in this paper.

5. Conclusions In the present study, a new approach for the Wiener–Hammerstein system identification was proposed and analysed focusing on the dissimilar laser joining of S.S 304 and ABS. A total number of seven datasets were employed in training and test phases for parameters such as pulse duration, frequency, current and power. Pulse duration, frequency, current and power were considered the input, and temperature was regarded as the output. In all tests, the welding speed was maintained constant at 6.2 mm/s with the focal length of 5 mm.

A Hammerstein-Wiener model was employed with 10-unit piecewise linear functions as the input and nonlinearity estimators as the output. To verify and check the predicted output, the identified Hammerstein-Wiener model was applied to the training dataset, indicating a good agreement between measured and predicted values. In addition, to check the fidelity of the method, performance criteria such as RMSE, MAPE and fitness were calculated.

The two main results of this study indicate that: • The acceptable fitness percentages of 93.92 in the training phase and 82.73 in the test phase are achieved.

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• The studied method proved to be accurate and reasonable enough to identify the laser welding process and other similar systems.

Since this is a problem-oriented paper rather than a mathematical-based one, it does not focus on the mathematical aspects of the model as well as the training method. Further research is needed in this regard.

Funding No funding was received for this work.

Declaration of Competing Interest No conflict of interest exists.

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