An Electrothermal Model Based Adaptive Control of Resistance Spot Welding Process

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1 Intelligent Control and utomation, 05, 6, Published Online May 05 in SiRes. n Eletrothermal Model Based daptive Control of Resistane Spot Welding Proess Ziyad Kas, Manohar Das Department of Eletrial and Computer Engineering, Oakland University, Rohester, US zrkas@oakland.edu, das@oakland.edu Reeived 6 February 05; aepted 8 May 05; published May 05 Copyright 05 by authors and Sientifi Researh Publishing In. his work is liensed under the Creative Commons ttribution International Liense (CC BY). bstrat Resistane Spot Welding (RSW) is a proess ommonly used for joining a stak of two or three metal sheets at desired spots. he weld is aomplished by holding the metalli workpiees together by applying pressure through the tips of a pair of eletrodes and then passing a strong eletri urrent for a short duration. Inonsistent weld and insuffiient nugget size are some of the ommon problems assoiated with RSW. o overome these problems, a new adaptive ontrol sheme is proposed in this paper. It is based on an eletrothermal dynamial model of the RSW proess, and utilizes the priniple of adaptive one-step-ahead ontrol. It is basially a traking ontroller that adjusts the weld urrent ontinuously to make sure that the temperature of the workpiees or the weld nugget traks a desired referene temperature profile. he proposed ontrol sheme is expeted to redue energy onsumption by 5% or more per weld, whih an result in signifiant energy savings for any appliation requiring a high volume of spot welds. he design steps are disussed in details. lso, results of some simulation studies are presented. Keywords Resistane Spot Welding, daptive Control, Nugget Formation, Energy Saving. Introdution In resistane spot welding, the welding proess begins by applying pressure on a stak of metal sheets, held together between a pair of eletrodes. weld urrent is then passed through the eletrodes, ausing resistive heating of the metal workpiees and the formation of a welded joint or nugget, as shown in Figure. he formation of a weld nugget strongly depends on the eletrial and thermal properties of the sheet and oating materials []. Sine the ontat resistane near the faying surfae is muh higher than the resistane of the sheets and eletrodes, most of the heating is onentrated near the faying surfae, ausing melting and formation of a nugget How to ite this paper: Kas, Z. and Das, M. (05) n Eletrothermal Model Based daptive Control of Resistane Spot Welding Proess. Intelligent Control and utomation, 6,

2 Figure. Resistane spot welding system. there. Depending on the thikness and type of material, welding urrent ranges from,000 to 0,000 amperes or more, while the voltage typially is between and 30 volts []. Resistane Spot Welding yle onsists of three main stages as follows: Stage : Squeeze time, whih is the time when eletrodes press the welded workpiees together. Stage : Weld time, whih is the time when welding urrent is applied produing heat at the faying surfae of the workpiees and thus reating a weld nugget. Stage 3: Hold time, whih is the time when eletrode fore still presses the workpiees together and ools the weld down after the welding urrent is swithed off. One of the most ommon appliations of resistane spot welding is in the automobile manufaturing industry, it is used almost universally to weld the sheet metals to form the ar body and parts. typial automotive vehile today requires about spot welds per vehile. Considering a worldwide annual prodution volume of 80 million automotive vehiles, an energy saving RSW ontroller an result in signifiant energy savings and redue arbon footprint aordingly. During the past two deades, a number of studies have been arried out to improve the RSW proess, whih fouses on monitoring and ontrol of weld parameters to improve weld quality. he RSW ontrol tehniques proposed to date inlude Proportional-Integral (PI) [3], Proportional-Derivative (PD) [4], Proportional-Integral- Derivative (PID) [5], Fuzzy [6]-[8], Neural Networks (NN) [9] [0], or a ombination of Fuzzy and NN []. he main drawbak of these tehniques is that they do not take into aount the thermal dynamis of the RSW proess, i.e. they do not utilize dynamial models that govern the heat transfer and nugget formation in the RSW proess. lso, these systems don t take into aount any welding proess variations, suh as variations in oating materials, eletrode degradation, and weld fore variations. In this paper, a novel approah to RSW ontrol is presented. his approah has not been explored by other researhers. We start with a simplified heat balane model of a RSW proess proposed in [] and [3], and then use it to design a ontroller. his thermal model of the heat balane is a funtion of nugget growth and it determines the temperature variation during welding time. his model is used later to design an adaptive-one-stepahead (OS) ontroller and an adaptive-weighted one-step-ahead (WOS) ontroller that ompensate for unknown proess variations and trak a desired referene temperature profile. Finally, some simulation results that show the performane of the proposed ontrollers are presented and ompared to the performane of a PID ontroller. Simulation results show that OS and WOS ontrollers are apable of traking a referene temperature profile when the weld parameters are unknown, as well as redue the energy needed to make a weld by 6%. he organization of this paper is as follows. Setion presents a simplified eletrothermal dynamial model of a RSW nugget formation proess. he design of adaptive OS and WOS ontrollers is disussed in Setion 3. Setion 4 presents the results of some simulation studies, and finally some onluding results are provided in Setion 5. 35

. Eletrothermal Dynamial Model of a RSW Nugget Formation Proess o start with, we onsider a simplified heat balane model of a RSW proess, presented in [3].

3 . Eletrothermal Dynamial Model of a RSW Nugget Formation Proess o start with, we onsider a simplified heat balane model of a RSW proess, presented in [3]. he simplified dynamial model of a RSW proess determines the heat balane in the system as a funtion of nugget temperature. For a simplified nugget model, shown in Figure, the heat balane an be desribed by the following equations: Q t is given by he total heat generation rate, g g = I( t ) denotes the welding urrent, and Q t I t R t (a) R( t) = Rw + R + Re (b) R t denotes the total resistane onsisting of the resistane of work piees, R w, ontat resistane, R, and eletrode resistane, R e. Sine R w and R e are very small ompared to the total ontat resistane R, R w and R e an be negleted in (b). he total ontat resistane an then be desribed as, eletrode-sheet sheet-sheet ( faying surfae) R = R t + R t () linear relationship between the resistane and temperature is assumed to model the heat generated as a funtion of temperature. hus, ρ R t eletrode-sheet sheet-sheet ( faying surfae) R t l = (d) ( ) p = ρ (e) ρ = ρ = ρ + αr θ θ (f) Figure. simplified model of a weld nugget. 36

4 ρ denotes the resistivity of the material, l denotes the distane from the melting interfae to eletrode ontat surfae, p denotes the penetration, is the ross setional area, ρ denotes the resistivity at referene temperature θ, θ and α r are the temperature to be ontrolled and the temperature oeffiient respetively. Substituting (f) in (d) and (e) we get Substituting (g) and (h) in (a) we get θ R t = t + (g) eletrode-sheet θ R t = t + (h) g sheet-sheet faying surfae 4 l ρα 3 4 = (i) l ρ ( αθ ) = r (j) pρα 3 = (k) pρ = ( αθ r ) (l) = ( θ + + 3θ + 4) Q t I t t t (m) θ = I t t + I t (n) 5 6 he heat of fusion required for nugget formation is given by: 5 = + 3 (o) 6 = + 4 (p) H f = H V (a) π n Vn = a p (b) H denotes the heat of fusion per unit volume, Vn denotes the nugget volume, and p, a denote the penetration and nugget radius respetively. Substituting (b) in (a) and normalizing over the weld duration, t, we get the heat of fusion per unit time: H f = Hap= t π Negleting the heat loss in the surroundings and the eletrodes, the heat required to raise temperature by dθ t is given by θ is the tempera- ρ denotes the density, ture rise. We rewrite (3a) as: p ρ dθ 7 () dq t = C t V (3a) p C denotes the speifi heat, V is the volume, and d ( t) dθ 8 dq t = t (3b) 37

5 he total heat loss rate is given by 8 ρc p πa p = (3) = + Q t Q t Q t (4a) L a r 0 θ t θ θ t βl b α = kπa + l kπa 0kπa βl kπa θ = + θ ( t) l b α l = θ t (4b) 9 0 kπa 0kπa β L 9 = + l b α (4) 0 kπa θ = l In the above equations, Q a ( t) and Q r t denote the axial and radial loss rates, respetively; k represents thermal ondutivity, a is the nugget radius; θ ( t), θ, represent the melting temperature and the interfae temperature at the work piee respetively; l is the distane from the melting interfae to the eletrodes ontat area; β represents the final penetration to work piee thikness ratio; L is the sheet thikness; b, α represent the eletrode radius and thermal diffusivity of work piee respetively. tt, + dt is given by he heat balane equation over time (4d) H f Q g ( t) = dt+ dq ( t) + Q L( t) dt t (5) Substituting (n), (), (3b), and (4b) in (5) and rearranging it, we get or, equivalently, ( t) dθ 8 = 5I ( t) θ( t) + 6I ( t) 9θ( t) (6a) dt dθ ( t) dt θ θ = I t t + I t t + (6b) 3 4 = 5 8 (6) = 6 8 (6b) 3 = 9 8 (6) = (6d)

6 For the sake of notational onveniene, let y( t) = θ ( t) and u( t) I ( t) =. hen (6b) an rewritten as dy t = u( t) y( t) + u( t) 3 y( t) + 4 (7) dt Equation (7) represents a bilinear eletrothermal dynamial model of a RSW proess. Note that this simplified model neglets the heat required to raise the temperature of the eletrodes and the nugget surroundings. lso, it assumes that most of the heating ours near the faying surfae due to its high ontat resistane. he size of the workpiees is assumed to be infinite in the radial diretion and the nugget shape is assumed to be a disk growing radially and axially in the same proportions. he nominal nugget diameter is assumed to be 4.5 L, L is the sheet thikness. Using a first order Euler approximation for d y with a sampling period s, the following disrete time equation is derived from the system Equation (7): dt y( k+ ) y( k) = u( k) y( k) + u( k) 3 y( k) + 4 (8a) or s ( ) y k+ = y k + Bu k + Cu k y k + D (8b) = (8) 3s B = (8d) s C= (8e) s D = (8f) 4 s lso, k denotes the disrete time index ( k = 0,,, ) and k s denote the sampling instanes. he above eletrothermal model is haraterized by four unknown parameters, namely,, B, C, and D. 3. Design of a RSW Controller o develop a ontrol sheme for ontrolling the nugget temperature of the RSW model presented by Equation (8a), we realize that it presents a bilinear system haraterized by some unknown parameters. hese parameters an vary from weld to weld, and in most ases we have no prior knowledge of the parameter values. In view of this, we propose to use an adaptive OS and WOS ontrollers. he proposed adaptive ontrol sheme involves measurement of the inputs and outputs of the system, estimation of unknown system parameters using a reursive least squares (RLS) parameter estimation algorithm, and omputation of a ontrol signal based on the estimated parameter values. lso, the temperature of the weld nugget is monitored indiretly by assuming it to be proportional to the ontat resistane. 3.. daptive OS and WOS Controllers In an adaptive ontroller, the sampled measurements, u( k ) and y k, are used to estimate the model parameters, BC,, and D in Equation (8b), using a reursive parameter estimation method, suh as reursive least square (RLS). he estimated values of these parameters are then used to ompute the OS/WOS ontrol signals. 3.. Parameter Estimation First we write model Equation (7) in the following form: ϕ y k+ = k X (9a) 39

7 ( k) y( k ) u( k ) y( k ) u( k ) ϕ = (9b) [ ] X B C D = (9) Next, the estimated value of θ is omputed reursively using the following RLS algorithm: P( k ) ϕ ( k ) ( k ) P( k ) κ( k ) ˆ θ k = ˆ θ k + y k ϕ k ˆ θ k ; k + ϕ ( ) P( k ) ( ) ϕ( ) ϕ( ) ( ) + ϕ( k ) P( k ) ϕ( k ) P k k k P k P k = [ γ ] (0a) (0b) X ˆ 0 = (0) P ( ) = σ I (0d) γ > 0 is a small number and σ > 0 is hosen to be large. lso, Ĉ( k ) is always onstrained to be non-negative, i.e., Given an estimate ˆX ( k ) of Cˆ k > ε > 0 for all k (0e) X, we define the predited output at time k + as: ϕ ˆ 3.3. daptive-one-step-head raking Controller yˆ k+ = k X k () One-step-ahead (OS) ontrol sheme for linear systems has been well investigated in [4]. n OS ontroller attempts to bring the predited output, y( k+ ) at time k +, to the desired value, y ( k+ ) in one step. hus, it minimizes the following ost funtion: J k y k y k ( + ) = ( + ) ( + ) he orresponding OS ontrol law is given by [4]: u k = ( ) B + Cy ( k ) y k + y k D he above ontrol signal needs to be onstrained by the imum urrent delivery apaity of the ontroller, u, as follows: u k, if 0, if u( k) ) u k < u k < u = 0, if 0 u u k u he adaptive OS ontroller uses the estimate, ˆX ( k ) in Equation () to ompute the ontrol signal, from the following adaptive version of Equation (3) above: u k k ˆ, Bk ˆ, Ck ˆ, and k. ( + ) Bˆ ( k) + Cˆ ( k) y( k) y k k ˆ yk Dk ˆ = () (3) (4) u k, (5) ˆD k denote the estimated values of BC,,, and D, respetively, at time 40

8 One of the potential drawbaks of OS ontrollers is exessive ontrol efforts that often result from attempt- y k+ in one step. o address this potential problem, an WOS ontroller is ing to bring y( k+ ) to disussed below daptive Weighted One-Step-head Controller he exessive effort to bring the output y( k+ ) to the desired value ( ) y k+ in one step using OS may result in an unfavorable saturation of the input. he adaptive weighted one-step-ahead ontroller attempts to seek a tradeoff between traking auray and ontrol effort by onsidering a slight generalization of the ost funtion () to the form (6) given below. hus, it minimizes the following ost funtion: λ J k y k y k u k ( + ) = ( + ) ( + ) +, 0< λ < is hosen to provide a desired tradeoff. he minimization of the ost funtion in (6) leads to the weighted one-step-ahead ontrol law [4]: u k = ( B+ Cy( k) )( y( k+ ) y( k) D) ( B + Cy ( k )) + λ he above ontrol law is also onstrained by the imum urrent delivery apaity, u, as shown in Equation (4) above. he hoie of λ provides a desired tradeoff between traking auray and ontrol effort. small λ results in good traking but requires high level of ontrol effort. large λ, on the other hand, redues ontrol efforts at the ost of traking auray. Xˆ k, in Equation () to ompute the ontrol signal, he adaptive WOS ontroller uses the estimate, u( k ) from the following adaptive version of Equation (7) above: u k k ˆ, Bk ˆ, Ck ˆ, and k. 4. Simulation Results and Disussion = ( Bˆ ( k) + Cˆ ( k) y( k) ) y( k+ ) ˆ ( k) y( k) Dˆ ( k) Bˆ ( k) + Cˆ ( k) y( k) + λ (6) (7) (8) ˆD k denote the estimated values of BC,,, and D, respetively, at time his setion presents the results of a simulation study showing the performane of the system with the proposed OS and WOS ontrollers and also ompare them with a PID ontroller. Eah ontroller is designed for traking a referene temperature profile. he referene temperature profile is a good indiator of the weld quality. herefore, it is desirable to keep the temperature variation lose to a desired variation urve, whih may be experimentally predetermined for the good welds. typial referene temperature profile for good weld is shown in Figure 3 below []. Basially, suh a urve is haraterized by a fast rise of temperature to melting point, melting of the workpiees at the faying surfae area whih auses a slight drop in temperature, followed by a ooling zone that results from removal of weld urrent. he atual nugget temperature is measured during the weld yle using the relationship desribed by Equation (f). Depending on the traking error signal, the welding urrent is adjusted so as to redue the temperature error. For these simulations, we have seleted two sheets of mild steel with the same thikness as the materials to be welded. he fore variation and eletrode wear are onsidered as unknown proess variables that impat the nugget size (diameter and penetration). he Figures below show the performane of the OS, WOS, and PID ontrollers due to 0% inrease in nugget diameter and 50% inrease in indentation from their desired values. 4

9 Figure 4 shows the performane of the OS ontroller using I = K, I denotes the imum urrent delivery apaity of the weld ontroller. We an see that the OS ontroller adapts to the parameter hange and fore the output temperature profile to follow the desired temperature profile. lso, we an see that the energy required for the weld is lower than that of the PID ontroller. Figure 5 and Figure 6 show the performane of WOS ontroller using I = K with λ = 0. and, respetively. Here we notie that when λ is high, the output temperature profile does not follow the desired output temperature profile well. However, inreasing λ results in dereasing the total energy required for the weld. Figure 7 shows the performane of the PID ontroller prior to any parameter hange using I = K. fter multiple trial and error attempts to get satisfatory results, the parameters of the PID ontrollers are: Proportional (P) = 0.5, Integral (I) = 6.56, Derivative (D) = 0. In Figure 8 we see that the PID ontroller looses trak of the referene temperature profile due to weld parameters hange. lso, we an see that PID ontroller requires more energy for the weld omparing to OS and WOS. Figure 3. Desired referene temperature profile. Figure 4. Performane of OS Controller with 0% inrease in nugget diameter and 50% inrease in indentation; I = K, Energy = 583 W. 4

10 Figure 5. Performane of WOS Controller with 0% inrease in nugget diameter and 50% inrease in indentation; λ = 0., I = K, Energy = 558 W. Figure 6. Performane of WOS Controller with 0% inrease in nugget diameter and 50% inrease in indentation; λ =, I = K, Energy = 470 W. Figure 7. Performane of PID Controller prior to unknown parameter variations; I = K, Energy 393 W =. 43

11 Figure 8. Performane of PID Controller with 0% inrease in nugget diameter and 50% inrease in indentation; K, Energy 63 W I = =. Comparing the simulation results for the three ontrollers, we an see that OS and WOS ontrollers ompensate for the parameter variations and trak the referene temperature profile quite well. Simulation results in Figure 5 for the WOS ontroller show satisfatory performane and a good tradeoff between traking error and total energy required for the weld regardless of hange in weld parameters. he output temperature profile follows the desired temperature profile reasonably well during the heating stage prior to the melting point. lso, we an see that the total energy required to make a weld using WOS is redued by 6% omparing to the PID ontroller when I = K. his an result in signifiant energy savings for appliations requiring a high volume of spot welds, suh as manufaturing of automotive vehiles. 5. Conlusion his paper presents a new approah for designing adaptive OS and WOS ontrollers for resistane spot welding proesses by utilizing a simplified eletrothermal dynamial model of the proess. Simulation results of OS and WOS performane are ompared with those of a PID ontroller. hese results indiate that using the proposed OS and WOS ontrollers, the nugget temperature profile is fored to trak a desired referene temperature profile in presene of unknown parameter variations. lso, these ontrollers redue the energy onsumed to perform a spot weld, whih an result in signifiant energy savings for appliations requiring a high volume of spot welds, suh as manufaturing of automotive vehiles. Referenes [] Zhang, H. and Senkara, J. (0) Resistane Welding Fundamentals and ppliations. aylor & Franis Group, Boa Raton. [] Govik,. (009) Modeling of the Resistane Spot Welding Proess. M.S. hesis, Institute of ehnology, Linkopings University, Linkoping. [3] Won, Y.J., Cho, H.S. and Lee, C.W. (983) Miroproessor-Based Control System for Resistane Spot Welding Proess. Proeedings of CC, San Franiso, -4 June 983, [4] Zhou, K. and Cai, L. (04) Nonlinear Current Control Method for Resistane Spot Welding. Proeedings of SME ransations on Mehatronis, 9, [5] Salem, M. and Brown, L.J. (0) Improved Consisteny of Resistane Spot Welding with ip Voltage Control. Proeedings of CCECE, Niagara Falls, 8- May 0, [6] Chen, X., raki, K. and Mizuno,. (997) Modeling and Fuzzy Control of the Resistane Spot Welding Proess. Proeedings of SICE, okushima, 9-3 July 997, [7] El-Banna, M., Filev, D. and Chinnam, R.B. (006) Intelligent Constant Current Control for Resistane Spot Welding. Proeedings of IEEE Conferene on Fuzzy Systems, Vanouver, 6- July 006,

12 [8] Chen, X. and raki, K. (997) Fuzzy daptive Proess Control of Resistane Spot Welding with a Current Referene Model. Proeedings of IEEE Conferene on Intelligent Proessing Systems, Beijing, 8-3 Otober 997, [9] Shriver, J., Peng, H. and Hu, S.J. (999) Control of Resistane Spot Welding. Proeedings of CC, San Diego, -4 June 999, [0] Ivezi, N., llen Jr, J.D. and Zaharia,. (999) Neural Network-Based Resistane Spot Welding Control and Quality Predition. Proeedings of IPMM, Honolulu, 0-5 July 999, [] Messler Jr, R.W., Jou, M. and Li, C.J. (995) n Intelligent Control System for Resistane Spot Welding Using a Neural Network and Fuzzy Logi. Proeeding of IC, Orlando, Otober 995, [] Kim, E.W. and Eagar,.W. (988) Parametri nalysis of Resistane Spot Welding Lobe Curve. SE ehnial Paper Series, Warrendale. [3] Kas, Z. and Das, M. (04) hermal Dynamial Model Based Control of Resistane Spot Welding. Proeedings of IEEE EI 04, Milwaukee, 5-7 June 04, [4] Goodwin, G.C. and Sin, K.S. (983) daptive Filtering Predition and Control. Prentie-Hall, Englewood Cliffs. 45

13 ppendix Boundedness of Nugget emperature Sine a RSW is a time limited proess ( t < 0.5 se usually), establishing a proof of asymptoti traking would be meaningless. However, it is important to make sure that the nugget temperature remains bounded during time ( 0, t). theoretial upper bound of the nugget temperature rise, θ ( t), during time, ( 0, t), an be established as follows. Notie the amount of heat absorbed = the amount of heat supplied the amount of heat loss Suppose θ 0, t hus, and C is a onstant. I denotes the imum weld urrent., C L is a onstant. hus, or, or, = rise in temperature during time, ( t) mount of heat obsorbed = C θ (9) mount of heat supplied = t I ( t) R( t) dt ti R (0a) 0 R = R + α θ (0b) r mount of heat lost = C θ () L C θ ti R + α θ C θ (a) r L ( r L) + (b) θ C ti α C ti R ti R θ C ti C αr + L whih proves the boundedness of the nugget temperature rise during weld time, ( t) 0,. () 46

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