Fig. 10 — Steadystate pool parameters. Fig. 11 — Open loop simulation result. Fig. 12 — Closed loop simulation result. (7) (8) ( ) L k NP Σ = 1 NI Σ ( ) C k NP = L C 1 NI where W(k – j), L(k – j), C(k – j) are the weld pool width, length, and convexity at instant k – j, NPW, NPL, and NPC are the orders for the previous measurements, and NIW, NIL, NIC are the orders for the input. aW(j), aL(j), aC(j) and bW(j), bL(j),bC(j) are the parameters for the ARMA models to be identified. The model orders are selected based on evaluating the model errors defined in Equations 1 and 2 and are visualized in Fig. 7, where RMSEs and average errors with respect to NI and NP for order from 1 to 20 are plotted. For the width, it is observed that when NPW is larger than 1, the differences in RMSE become negligible. In this sense, NPW = 1 was selected. Similarly, the order of the input can be determined. For the weld pool length and convexity, an identical procedure can be employed, and the full set of model orders is listed in Table 2. Modeling Result and Analysis The least squares algorithm (Ref. 16) is utilized to identify the model parameters in Equations 6–8. The identified model parameters are listed in Table 3. The modeling results are plotted in Fig. 8 and the model errors are shown in Table 4. It was observed that the identified model can correlate the weld pool parameters to the system input with acceptable accuracy. From Fig. 8 it is shown that the proposed ARMA model is able to model the fluctuating weld pool parameters. To verify the proposed model, two verification experiments were conducted, and the results are shown in Fig. 9. It is shown in Fig. 9B–D that the ARMA models can estimate the weld pool parameters with acceptable accuracy. Steady-state models for the width, length, and convexity can be derived from Equations 6 to 8 and are expressed as (9) (10) (11) Figure 10 plots the steady state width, length, and convexity for welding speeds from 0.6 to 1.5 mm/s. It was observed that as the welding speed increased, the width, length, and convexity all decreased. This is understandable because the increase in welding speed significantly decreases the heat input, which is a major influence on the weld pool surface geometry. A decreased heat input causes the decrease in the pool width and length. Because the amount of metal melted is also decreased, the convexity tends to decrease as well. Model Predictive Control Algorithm Model predictive control (MPC) has received considerable attention in past decades in both theoretical developments and applications in industrial practices (Refs. 17–20). Recently, several researchers have developed nonlinear model predictive control (NMPC) algorithms (Refs. 21–23). Σ Σ ( ) ( ) ( ) ( ) = − + − = a jC k j b j u k j j C j 1 C C ( ) ( ) ( ) ( ) = − + − = a j L k j b j u k j j L j 1 L L = 5.6274 W S s = 6.6846 L S s = 0.1292 C S s WELDING RESEARCH 130-s WELDING JOURNAL / APRIL 2015, VOL. 94
Welding Journal | April 2015
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