Fig. 13 — Experiment 1 results. A — Welding current and weld pool parameters; B — control input; C — frontside bead; D — backside bead. However, most of the above methods may require tremendous computational effort for nonlinear optimization, thus may not be most desirable for online industrial applications. Following, an analytical solution to the proposed ARMA model predictive control algorithm is derived. MPC Algorithm At instant k, the controller needs to determine the control action u(k) based on the feedback/current 3D weld pool characteristic parameters to drive the pool parameters to their desired values. In a predictive control, prediction equations should be first developed to predict the outputs based on the input. According to Equations 6–8, the system incremental model can be expressed as (12) ( )= ( −1) + − =1 W k a W k 3 W One-step-ahead prediction of the weld pool width yields (13) ( ) ( ) W k a W k 3 =1 3 W b ju k j a W k W Similarly, one-step-ahead predictions of the length and convexity are expressed as (14) (15) ( +1 ) = ( ) L k a L k 5 L + +1− =2 b ju k j L j L ( +1 ) = ( ) C k a C k 5 + +1− =2 b ju k j In order to achieve a desirable control, the following cost function must be minimized (16) where y*(k + 1) OEW*(k + 1) L*(k + 1) C*(k + 1) represents the desired setpoint for one-step prediction, plus h1 and h2 are the relative coefficients for the length and convexity, respectively. In this study, it was chosen based on the pool parameters’ importance relative to the weld penetration specified in Ref. 16. wb = 1.79W–0.57L–10.8C–0.99 (17) h1 =0.57/1.79 = 0.32 and h2 = 10.8/1.79 = 6.03 were chosen. The desired set-point for one-step prediction should be designed depending on the application being adressed. In this study, the desired one-step setpoint y*(k + 1) is defined as y*(k+1) = gy(k)+(1–g)yc(k+1) (18) where yc(k) represents the desired weld pool parameters (width, length, and convexity), and gOE(0,1) is the smoothing coefficient. As g becomes larger, the system tracks the set-point with slower speed but better robustness and smoothness. In this study, g = 0.5 achieved an appropriate tradeoff between response speed and robustness, and was selected. The fluctuations in the input generated nonsmooth welds, which was not desirable in our application. Thus, energetic control actions must be avoid- Σ b W ( ju ) ( k j ) j Σ Σ ( ) ( ) ( ) ( ) ( ) ( ) ( ) +1 = + +1− = + +1− + 1 =2 b ju k j b u k j W W j W Σ ( ) ( ) ( ) ( ) = 1 b u k Σ ( ) ( ) ( ) ( ) + 1 b u k C C j C ( ) ( ) ( ) ∗ 2 ( ) = ( +1− ) ( +1 ) J k,u W k W k ∗ 2 ( ) ( ) ( ) ( ) +η +1 − +1 1 L k L k +η +1 − +1 2 ∗ 2 C k C k WELDING RESEARCH APRIL 2015 / WELDING JOURNAL 131-s A B C D
Welding Journal | April 2015
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