In resistance spot welding, the welding process begins by applying pressure on a stack of metal sheets, held together between a pair of electrodes. A weld current is then passed through the electrodes, causing resistive heating of the metal workpieces and the formation of a welded joint or nugget, as shown in Figure 1. The formation of a weld nugget strongly depends on the electrical and thermal properties of the sheet and coating materials [1] . Since the contact resistance near the faying surface is much higher than the resistance of the sheets and electrodes, most of the heating is concentrated near the faying surface, causing melting and formation of a nugget

there. Depending on the thickness and type of material, welding current ranges from 1,000 to 20,000 amperes or more, while the voltage typically is between 1 and 30 volts [2] .

A Resistance Spot Welding cycle consists of three main stages as follows:

Stage 1: Squeeze time, which is the time when electrodes press the welded workpieces together.

Stage 2: Weld time, which is the time when welding current is applied producing heat at the faying surface of the workpieces and thus creating a weld nugget.

Stage 3: Hold time, which is the time when electrode force still presses the workpieces together and cools the weld down after the welding current is switched off.

One of the most common applications of resistance spot welding is in the automobile manufacturing industry, where it is used almost universally to weld the sheet metals to form the car body and parts. A typical automotive vehicle today requires about 4000 - 6000 spot welds per vehicle. Considering a worldwide annual production volume of 80 million automotive vehicles, an energy saving RSW controller can result in significant energy savings and reduce carbon footprint accordingly.

During the past two decades, a number of studies have been carried out to improve the RSW process, which focuses on monitoring and control of weld parameters to improve weld quality. The RSW control techniques proposed to date include Proportional-Integral (PI) [3] , Proportional-Derivative (PD) [4] , Proportional-Integral- Derivative (PID) [5] , Fuzzy [6] - [8] , Neural Networks (NN) [9] [10] , or a combination of Fuzzy and NN [11] . The main drawback of these techniques is that they do not take into account the thermal dynamics of the RSW process, i.e. they do not utilize dynamical models that govern the heat transfer and nugget formation in the RSW process. Also, these systems don’t take into account any welding process variations, such as variations in coating materials, electrode degradation, and weld force variations.

In this paper, a novel approach to RSW control is presented. This approach has not been explored by other researchers. We start with a simplified heat balance model of a RSW process proposed in [12] and [13] , and then use it to design a controller. This thermal model of the heat balance is a function of nugget growth and it determines the temperature variation during welding time. This model is used later to design an adaptive-one-step- ahead (AOSA) controller and an adaptive-weighted one-step-ahead (AWOSA) controller that compensate for unknown process variations and track a desired reference temperature profile. Finally, some simulation results that show the performance of the proposed controllers are presented and compared to the performance of a PID controller. Simulation results show that AOSA and AWOSA controllers are capable of tracking a reference temperature profile when the weld parameters are unknown, as well as reduce the energy needed to make a weld by 6%.

The organization of this paper is as follows. Section 2 presents a simplified electrothermal dynamical model of a RSW nugget formation process. The design of adaptive OSA and WOSA controllers is discussed in Section 3. Section 4 presents the results of some simulation studies, and finally some concluding results are provided in Section 5.

To start with, we consider a simplified heat balance model of a RSW process, presented in [13] . The simplified dynamical model of a RSW process determines the heat balance in the system as a function of nugget temperature. For a simplified nugget model, shown in Figure 2, the heat balance can be described by the following equations:

The total heat generation rate, is given by

where denotes the welding current, and denotes the total resistance consisting of the resistance of work pieces, , contact resistance, , and electrode resistance,. Since and are very small compared to the total contact resistance, and can be neglected in (1b).

The total contact resistance can then be described as,

A linear relationship between the resistance and temperature is assumed to model the heat generated as a function of temperature. Thus,

where denotes the resistivity of the material, denotes the distance from the melting interface to electrode contact surface, p denotes the penetration, A is the cross sectional area, denotes the resistivity at reference temperature, and are the temperature to be controlled and the temperature coefficient respectively.

Substituting (1f) in (1d) and (1e) we get

where

Substituting (1g) and (1h) in (1a) we get

where

The heat of fusion required for nugget formation is given by:

where H denotes the heat of fusion per unit volume, denotes the nugget volume, and p, a denote the penetration and nugget radius respectively. Substituting (2b) in (2a) and normalizing over the weld duration, , we get the heat of fusion per unit time:

Neglecting the heat loss in the surroundings and the electrodes, the heat required to raise temperature by is given by

where denotes the density, denotes the specific heat, is the volume, and is the temperature rise. We rewrite (3a) as:

where

The total heat loss rate is given by

where

In the above equations, and denote the axial and radial loss rates, respectively; repre-

sents thermal conductivity, a is the nugget radius;, , represent the melting temperature and the interface temperature at the work piece respectively; is the distance from the melting interface to the electrodes contact area; represents the final penetration to work piece thickness ratio; L is the sheet thickness; represent the electrode radius and thermal diffusivity of work piece respectively.

The heat balance equation over time is given by

Substituting (1n), (2c), (3b), and (4b) in (5) and rearranging it, we get

or, equivalently,

where

For the sake of notational convenience, let and. Then (6b) can rewritten as

Equation (7) represents a bilinear electrothermal dynamical model of a RSW process. Note that this simplified model neglects the heat required to raise the temperature of the electrodes and the nugget surroundings. Also, it assumes that most of the heating occurs near the faying surface due to its high contact resistance. The size of the workpieces is assumed to be infinite in the radial direction and the nugget shape is assumed to be a disk growing radially and axially in the same proportions. The nominal nugget diameter is assumed to be, where L is the sheet thickness.

Using a first order Euler approximation for with a sampling period, the following discrete time equation is derived from the system Equation (7):

or

where

Also, k denotes the discrete time index and denote the sampling instances. The above electrothermal model is characterized by four unknown parameters, namely, A, B, C, and D.

To develop a control scheme for controlling the nugget temperature of the RSW model presented by Equation (8a), we realize that it presents a bilinear system characterized by some unknown parameters. These parameters can vary from weld to weld, and in most cases we have no prior knowledge of the parameter values. In view of this, we propose to use an adaptive OSA and WOSA controllers.

The proposed adaptive control scheme involves measurement of the inputs and outputs of the system, estimation of unknown system parameters using a recursive least squares (RLS) parameter estimation algorithm, and computation of a control signal based on the estimated parameter values. Also, the temperature of the weld nugget is monitored indirectly by assuming it to be proportional to the contact resistance.

This section presents the results of a simulation study showing the performance of the system with the proposed AOSA and AWOSA controllers and also compare them with a PID controller. Each controller is designed for tracking a reference temperature profile.

The reference temperature profile is a good indicator of the weld quality. Therefore, it is desirable to keep the temperature variation close to a desired variation curve, which may be experimentally predetermined for the good welds. A typical reference temperature profile for good weld is shown in Figure 3 below [1] . Basically, such a curve is characterized by a fast rise of temperature to melting point, melting of the workpieces at the faying surface area which causes a slight drop in temperature, followed by a cooling zone that results from removal of weld current. The actual nugget temperature is measured during the weld cycle using the relationship described by Equation (1f). Depending on the tracking error signal, the welding current is adjusted so as to reduce the temperature error.

For these simulations, we have selected two sheets of mild steel with the same thickness as the materials to be welded. The force variation and electrode wear are considered as unknown process variables that impact the nugget size (diameter and penetration). The Figures below show the performance of the AOSA, AWOSA, and PID controllers due to 20% increase in nugget diameter and 50% increase in indentation from their desired values.

Figure 4 shows the performance of the AOSA controller using, where denotes the maximum current delivery capacity of the weld controller. We can see that the AOSA controller adapts to the parameter change and force the output temperature profile to follow the desired temperature profile. Also, we can see that the energy required for the weld is lower than that of the PID controller.

Figure 5 and Figure 6 show the performance of AWOSA controller using with and

1, respectively. Here we notice that when is high, the output temperature profile does not follow the desired output temperature profile well. However, increasing results in decreasing the total energy required for the weld.

Figure 7 shows the performance of the PID controller prior to any parameter change using. After multiple trial and error attempts to get satisfactory results, the parameters of the PID controllers are: Proportional (P) = 0.5, Integral (I) = 26.56, Derivative (D) = 0.

In Figure 8 we see that the PID controller looses track of the reference temperature profile due to weld parameters change. Also, we can see that PID controller requires more energy for the weld comparing to AOSA and AWOSA.

Comparing the simulation results for the three controllers, we can see that AOSA and AWOSA controllers compensate for the parameter variations and track the reference temperature profile quite well. Simulation results in Figure 5 for the AWOSA controller show satisfactory performance and a good tradeoff between tracking error and total energy required for the weld regardless of change in weld parameters. The output temperature profile follows the desired temperature profile reasonably well during the heating stage prior to the melting point. Also, we can see that the total energy required to make a weld using AWOSA is reduced by 6% comparing to the PID controller when. This can result in significant energy savings for applications requiring a high volume of spot welds, such as manufacturing of automotive vehicles.

This paper presents a new approach for designing adaptive OSA and WOSA controllers for resistance spot welding processes by utilizing a simplified electrothermal dynamical model of the process. Simulation results of AOSA and AWOSA performance are compared with those of a PID controller. These results indicate that using the proposed AOSA and AWOSA controllers, the nugget temperature profile is forced to track a desired reference temperature profile in presence of unknown parameter variations. Also, these controllers reduce the energy consumed to perform a spot weld, which can result in significant energy savings for applications requiring a high volume of spot welds, such as manufacturing of automotive vehicles.

Since a RSW is a time limited process (usually), establishing a proof of asymptotic tracking would be meaningless. However, it is important to make sure that the nugget temperature remains bounded during time. A theoretical upper bound of the nugget temperature rise, during time, can be established as follows.

Notice the amount of heat absorbed = the amount of heat supplied ? the amount of heat loss

Suppose

= rise in temperature during time,

Thus,

where is a constant.

where

and denotes the maximum weld current.

where, is a constant.

Thus,

or,

or,

which proves the boundedness of the nugget temperature rise during weld time,