A B C D Fig. 14 — Experiment 2 results. A — Welding current and weld pool parameters; B — control input; C — frontside bead; D — backside bead. ed. The modified following cost function was used to penalize changes in control (19) ( ) = ( +1− ) ( +1 ) J k,u W k W k ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) +η +1 − +1 L k L k +η +1 − +1 +λ − −1 λ − C k C k u k u k u k u k where l1 is the penalty weight on the control change, which can be determined based on the scale of the static gain to the output. In this study, l1 = 52 (mm2/mm/s) was chosen. This implies that an error of 1 mm in the width has the same contribution to the cost function as the input change of 5/÷mm/s. l2 is the penalty weight on the control signal deviation from the expected welding speed value u*(k), which represents another constraint on the speed control based on authors’ previous study on human welder’s intelligence learning and control (Ref. 10). In this study l2 = 12 (mm2/mm/s) is chosen. The control law was calculated such that J(k,u)/u = 0 (20) The control signal was finally expressed as u(k) = –N/D (21) where (22) 2 D b b 1 W L C = 1 + ⎢⎢⎢⎢⎢⎢ ⎥⎥⎥⎥⎥⎥ ⎢⎢⎢⎢⎢⎢ ⎥⎥⎥⎥⎥⎥ j ⎢⎢⎢⎢⎢⎢ ⎥⎥⎥⎥⎥⎥ W 3 =2 ∗ 1 j L 5 =2 ∗ C 5 =2 ∗ W W L L 2 C C j (23) Since the control input is defined as u = 1/√S, the optimal welding speed, i.e., Super Welder, can thus be conveniently calculated by S* = 1/u2. It is apparent that Equation 21 provides a closed form to the solution and no online optimization is required. Thus, the obtained control algorithm can be easily incorporated into real-time welder training applications. Controller Simulation Open-loop control was first simulated by employing Equations 6–8, and the result is plotted in Fig. 11. From 0 to 20 s, the welding speed was set at 1.4 mm/s, and the weld pool width, length, and convexity was 4.8, 5.8, and 0.1 mm, respectively. From 20 to 40 s, the welding speed was changed to 1.2 mm/s. The weld pool parameters gradually reached different steady-state values. The settling time was about 4 s for the width, 5 s for the length, and 3 s for the convexity. For 40 to 60 s, 60 to 80 s, and 80 to 100 s, the welding speed was set to 1.0, 0.8, and 0.6 mm/s, respectively. With no closedloop control, the weld pool parameters cannot be maintained at their desired values. Figure 12 shows the simulation result with the proposed closed-loop control algorithm. From 0 to 15 s is the open-loop control period, with the ( ) ( ) ( ) ( ) ( ) ∗ 2 1 ∗ 2 2 ∗ 2 1 2 2 ∗ 2 ( ) ( ) ( ) = 1+η 1 +η 1 +λ +λ 2 2 2 1 2 b Σ Σ Σ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) +1 − +1 ⎡ ⎣ ⎤ ⎦ +η 1 + +1 − +1 ⎡ ⎣ ⎤ ⎦ +η + +1 − +1 ⎡ ⎣ ⎤ ⎦ −λ − −λ 1 2 ∗ N b a W k b ju k – j W k b a L k b j u k – j L k b 1 a C k b j u k – j C k u k 1 u k WELDING RESEARCH 132-s WELDING JOURNAL / APRIL 2015, VOL. 94
Welding Journal | April 2015
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