A Model-Free Adaptive Control of Plsed GTAW F.L. Lv 1, S.B. Chen 1, and S.W. Dai 1 Institte of Welding Technology, Shanghai Jiao Tong University, Shanghai 00030, P.R. China Department of Atomatic Control, NAEI, Yantai, 64001 arry@sjt.ed.cn Abastract. A closed-loop system is developed to control the weld pool geometry, which is specified by the bacside weld width. The welding crrent and speed are selected as the control variables. A MISO ARX model is sed to model this dynamic process. Becase of ncertainties in the GTAW process, it is very difficlt to design an effective control scheme by conventional modeling and control methods. A model-free adaptive control algorithm has been developed to control the welding process. Simlation confirmed that the developed control system is effective in achieving the desired fsion state. 1 Introdction The dynamic identification and control of arc welding processes has been explored throgh a nmber of stdies. Advanced control techniqes sch as adaptive control have been sed to generate sond welds [1,]. Artificial intelligence methodology was developed for modeling and controlling the welding process. Nmeros exciting controls with perfect performance have been achieved. George E. Coo[3] stdied the application for the variable polarity plasma arc welding. A self-learning fzzy neral networ control system of topside width enabled adaptive altering of welding parameters to compensate for changing environments in Chen S.B. s stdy[4]. Y M. Zhang[5] sed a nerofzzy model to model the welding process. Based on the dynamic fzzy model, a predictive control system has been developed to control the welding process. Zhang G.J.[6] investigated feasible intelligent control schemes. A doble-variable controller was designed. In fact, it jst is a spervisory control, not considering the cople relationship of variables. Recently, a model-free adaptive control scheme has been explored, which based on information from the I/O data of the controlled plant[7,8]. The main objective of this research was to explore applications for mode-free adaptive control algorithm in arc welding control, and in particlar the GTAW MISO(many inpt simple otpt process. The organization of this paper is as follows: we first identify the doble inpt simple otpt dynamic model of the GTAW process for simlation of the GTAW process; the basic model-free adaptive control algorithm is presented; finally, the simlation of GTAW process is shown. T.-J. Tarn et al. (Eds.: Robot. Weld., Intellige. & Atomation, LNCIS 36, pp. 333 339, 007. springerlin.com Springer-Verlag Berlin Heidelberg 007
334 F.L. LY, S.B. Chen, and S.W. Dai GTAW Process Modeling GTAW process is controlled by a nmber of parameters, inclding the welding crrent, arc length, and welding speed. As we all now, the correlation between the weld pool and arc length may not be straightforward. Compared with the arc length, the roles of the welding crrent and welding speed in determining the weld pool and weld fsion geometry are mch more significant and definite. Ths we select the welding speed (v and the welding crrent (I as control variables. The controlled process can therefore be defined as a GTAW process in which the welding crrent and speed are adjsted on-line to achieve the desired bacside width (w b of the weld pool. A polynomial AtoRegressive with exogenos inpt or ARX model representation[9] is selected as the model representation in this stdy. For a mltivariable system with one otpt and m inpts, consider the ARX model below: 1 1 Aq ( y ( = B( q ( + e ( (1 The model belongs to linear-in-the-parameter model, therefore, the parameter estimation can be performed sing least sqare method. The model in eqation (1 can be represented y ( = Τ ( + e ( The identification is ths simplified by estimation the model parameters. There are n+nm parameters to be identified, and here m is the nmber of inpts which is welding crrent (I and welding speed (v respectively, so 1 is welding crrent, is welding speed, and m eqals. Thogh experiment data, we can determine approximately that the evalation of n is 5. ARX model of bacside weld width (w b with welding parameters (I and v is derived sing the least sqare method developed with the Matlab program. First we achieve fll parameter model, then a redced model can be derived sing statistic hypothesis testing method as follows: y ( = ay ( 1 + ay ( + ay ( 3 + ay ( 4 1 3 4 + ay ( 5 + b( 1 + b( + b( 4 5 11 1 1 1 14 1 + b ( 1 + b ( 3 + b ( 5 1 3 5 where A= [ a1 a a3 a4 a5] = [ 1.45-0.7935 0.4569-0.314 0.11518] B = [ b11 b1 b14 b1 b3 b5 ] = [ 0.0085696-0.3748 0.0039714-0.1686 0.003674-0.079501], y is the bacside weld width, 1 and are the welding crrent and the welding speed, respectively. The feasibility of this model is verified by comparing the simlation reslts with the Matlab program and actal otpts. The sqare sm of residals is 0.0303437.3 model-free adaptive control design (
A Model-Free Adaptive Control of Plsed GTAW 335.1 Universal Process Model The following general discrete MISO nonlinear systems is considered = 0,1,, ( 1 ( n m ( 1 y ( + 1 = f Y, (, U, + 1 Where stands for discrete time y+ represents a one-dimensional state otpt is an inpt variable Y n = { y(,, y( n} (3 are the sets of system otpts m U = { (,, ( m} n and m are the orders of otpt y ( and inpt ( f ( is a general nonlinear fnction are the sets of system inpts Following assmptions are made abot the controlled plant: 1. When the system is in the steady state, it satisfies the condition that if ( = ( 1, then y ( + 1 = y (. The nonlinear fnction f ( has a continos gradient with respect to control inpt (. From the assmptions above, we have ( n m 1 1 1 ( n m 1 y ( + 1 y ( = f Y, (, U, + 1 f Y, ( 1, U, n m n m (, (, 1, 1 (, ( 1,, 1 ( n, ( 1, m ( 1 1 1, 1 n 1, ( 1, m, = f Y U + f Y U + + f Y U + f Y U Let n m n 1 m 1 ( ( 1 f Y, ( 1, U, + 1 f Y, ( 1, U, = ξ ( + 1 And sing the mean vale theorem in the Calcls, we obtain n m n m (, (, 1, + 1 (, ( 1,, + 1 (, ( 1,, 1 Τ n m = f Y 1 [ ( ( 1 U + ] f Y U f Y U
336 F.L. LY, S.B. Chen, and S.W. Dai where ( -1 = ( -1 + θ ( ( ( -1 we have, θ satisfies 0 θ 1. Therefore, ( 1 Τ ( [ ] y ( + 1 y ( = f ( -1, + 1 ( ( -1 + ξ ( + 1 (4 If ( ( -1 0, let ( ( ( = ( -1 f ( -1, 1 ( ( -1 ( ( -1 + + ξ ( + 1 Then eqation (4 can be written as [ ] Τ y ( + 1 y ( = ( ( ( -1 (5 where ( can be considered a psedo gradient of model (5. Note that when the system is in a steady state, becase of ( ( - 1 = 0, we have y ( + 1 = y (, so in this case, (5 is a valid expression. Eqation (5 is called niversal model.. Model-Free Adaptive Control Algorithm (..1 Estimation of the Psedo Gradient It is clear that the necessary condition that the niversal model (5 can be sed in practice is that the estimation of (, denoted as (, is available in real-time, and is sfficiently accrate. Considering the control action is nown, define the cost fnction Τ J( ( = y ( + 1 y( ( ( 1 + µ ( ( 1 (6 where ( -1 = ( -1- ( -, becase at the moment ( is naccessible we sbstitte ( 1 for it, y ( + 1 is the desired otpt of the controlled plant, µ is positive weighting constant which constrains the change of the psedo gradient ( ( 1. By sing (5, the minimization of the cost fnction (6 gives estimation ( η µ ˆ( = ˆ( 1 + ( 1 + ( 1 (7
A Model-Free Adaptive Control of Plsed GTAW 337 ( y ( ˆ ( 1 ( 1 where η is a sitable small positive nmber... Design of Model-Free Adaptive Control The model-free adaptive control is described as follows: Assme that the observed data { ( 1, y( } ( = 1, are nown, and the expected otpt y ( + 1 at (+1th time is given. Find a controller (, sch that the otpt of the system y+ ( 1 matches y ( + 1. In order to achieve a robst control, it is reqired that the following cost fnction is minimized: J( ( = y ( + 1 y( + 1 + λ ( ( 1 where λ is the weight. The analytic soltion is ( ( ρ λ ( ( = ( 1 + ( + ( y ( + 1 y( (8 where ρ is called a control parameter, which selection is closely related to the convergence of the control law. Eqation (8 is called the basic form of the model-free adaptive control law. 3 Simlation of GTAW Process Having analyzed the GTAW process and developed the GTAW MISO model (, we will demonstrate the effectiveness of the model-free adaptive control algorithm developed in GTAW process. Following are several steps that describe how the model-free adaptive algorithm wors. 1. For the observed inpt otpt data { ( 1, y( } and the developed niversal model (5, we can obtain ( which is the estimation of psedo partial derivative (, sing least sqares algorithm.. For the desired otpt y ( + 1, we have ( ( ρ λ ( ( = ( 1 + ( + ( y ( + 1 y( Then a new set of data { ( 1, y( } can be achieved, applying the control action ( to the GTAW process (here we se the model (.
338 F.L. LY, S.B. Chen, and S.W. Dai 3. Repeat steps 1 and above to generate a serial of data { ( 1, y( }. The desired bacside weld width was chosen as y = 5 mm. On selection the initial estimation of psedo partial derivative as = [ ] (0 0.5 0. Τ, Fig. shows the convergence procedre of the psedo partial derivative. Fig.3 and Fig.4 show the response of the system nder the model-free adaptive control with η = 1, µ = 1, ρ = 1 and λ = 5. Fig. 1. Psedo partial derivative Fig.. Control inpts Fig. 3. Bacside weld width
A Model-Free Adaptive Control of Plsed GTAW 339 4 Conclsion Becase arc welding is characterized as inherently variable, nonlinear, time varying and having a copling among parameters. Also the variations in the welding conditions case ncertainties in the dynamics. So it is very difficlt to design an effective control scheme by conventional modeling and control methods. It is has been shown by simlation that this process has been sccessflly controlled sing the model-free adaptive algorithm. Meanwhile this control method only needs the observed inpt otpt data, ths the MISO GTAW model developed is jst sed to simlate. Ths, the developed model-free adaptive control provides a promising technology for GTAW qality control. Acnowledgement This wor is partially spported by the National Natral Science Fondation of China nder Grand No. 60474036 and the Key Fondation Program of Shanghai Sciences & Technology Committee nder Grant No. -&. References 1. Kovacevic R., Zhang Y.M., (1997 Real-time image processing for monitoring the free weld pool srface. ASME Jornal of Mnfactring Science and Engineering 119(5:161-169.. Song J.B., Hardt D.E. (1994 Dynamic modeling and adaptive control of the gas metal arc welding process. ASME Jornal of Dynamics Systems, Measrement, and Control 116(3:405-410. 3. George E. Coo, Robert Joel Barnett, Kristinn Andersen, Alvin M. Strass (1995 Weld modeling and control sing artificial neral networs. IEEE Transactions on Indstry Applications 31(6:1484-1491. 4. Chen S.B., W L., Wang Q.L., Li Y.C. (1997 Self-learning fzzy neral networ and compter vision for control of plsed GTAW. Welding Jornal 76(5:01s-09s. 5. Y M. Zhang, Radovan Kovacevic (1998 Nerofzzy model-based predictive control of weld fsion zone geometry. IEEE Transactions on Fzzy Systems 6(3:389-401. 6. Zhang G.J., Chen S.B., W L. (005 Intelligent control of plsed GTAW with filter metal. Welding Jornal 84(1:9s-16s. 7. Zhongsheng Ho, Wenh Hang (1997 The model-free learning adaptive control of a class of SISO nonlinear systems. Proceedings of The American Control Conference 343-344. 8. Zhi-Gang Han, Xingho Y (004 An adaptive model free control design and its applications. Indstrial Informatics, 004.INDIN 04. 004 nd IEEE International Conference on 43-47. 9. Li Yanjn, Zhang Ke (005 Xitong bianshi liln ji yingyong. Gangfang Gongye Ch Ban She.