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Iterative Learning Control of the Cooling Rate in a Dual-Laser Powder Bed Fusion Process

Lauren Bogo, Adam Clare, Dominic Liao-McPherson

arXiv:2609.28734v1eess.SY

TL;DR

LPBF thermal history governs solidification microstructure, while dual-laser parameter calibration is challenging and existing feedforward approaches require repeated tuning or substantial evaluation budgets. The paper combines model-based search directions with iterative plant feedback for dual-laser process control. In simulation, the controller reaches a tight band of low-cost plant solutions across seeds, while model-only optimization converges on its surrogate but remains at higher plant cost.

  • Problem

    LPBF microstructure and mechanical properties depend on thermal history, but calibrating dual-laser settings is challenging and feedforward tuning must be repeated across materials, machines, or geometries.

    Source: §I · p. 1

  • Method

    OB-ILC optimizes laser powers, lag distance, and scan velocity using simulation-derived search directions and plant feedback to correct model mismatch.

    Source: §I · p. 2

  • Results

    Across 7 initial conditions, ILC reaches a tight band of plant solutions with CV 5.6% within roughly 20 iterations, while model-only optimization overshoots the target cooling profile on the plant.

    Source: B. Convergence history at a single seed · p. 5, §VII · p. 7

  • Takeaways & Limitations

    The framework requires no algorithmic changes to use pyrometer feedback on hardware, though experimental validation remains future work.

    Source: §VII · p. 7

    Validation uses thermal simulations, with experimental validation left for future work.

    Source: §I · p. 2, §VII · p. 7

Abstract

from arXiv · show

The thermal history of the melt pool in laser powder bed fusion (LPBF) additive manufacturing processes governs the solidification microstructure and the mechanical properties of the resulting 3D-printed parts. Dual-laser systems offer additional degrees of freedom to control the cooling profile by reheating material behind the melt pool, but calibrating process parameters is challenging due to the complex physics of the process. We present an optimization-based iterative learning controller that determines optimal power, velocity, and offset settings by judiciously combining simulations and experiments: the model supplies search directions while feedback obtained from experiments on the real plant corrects for parameter errors, enabling convergence despite model inaccuracies. The approach is validated in simulation using a high-fidelity thermal model as a plant surrogate, with deliberate mismatches in absorption coefficient, latent heat treatment, and powder-bed effective conductivity between plant and model. Results show that the controller drives the plant cost down by over an order of magnitude and reaches a tight band of low-cost solutions across seeds, while model-only feedforward optimization stalls at a substantially higher plant cost despite appearing to converge on the surrogate.

I. INTRODUCTION

LPBF parts can have columnar grains, anisotropic properties, and residual stress, making post-processing costly; dual-laser reheating offers a way to modify solidification cooling. The paper introduces an optimization-based iterative learning controller that combines model-derived search directions with plant feedback to tune dual-laser parameters.

I. INTRODUCTION

  • LPBF parts’ columnar grains, anisotropic properties, and residual stress can require costly post-processing that offsets additive manufacturing’s efficiency gains.The introduction argues that controlling microstructure during the build could improve efficiency and reduce cost.Source: §I · p. 1
  • Dual-laser LPBF studies had used feedforward parameter sweeps or simulation-based choices, which must be repeated for each material, machine, or geometry.The passage also notes that derivative-free alternatives scale poorly with input dimension.Source: §I · p. 1
  • The controller drives plant cost down by over an order of magnitude within roughly 20–40 plant evaluations, using model gradients for search directions and plant feedback to correct model error.The approach targets a calibration burden that makes grid search expensive and Bayesian optimization demanding at four input dimensions.Source: §I · pp. 1–2
  • OB-ILC optimizes leading and trailing laser powers, lag distance, and scan velocity to track target thermal profiles through solidification.A differentiable simulation supplies search directions, while experiments on the plant correct model mismatch at each iteration.Source: §I · p. 2
  • The work presents the first application of ILC to dual-laser thermal-profile optimization in LPBF, extending prior ILC applications in additive manufacturing.Prior applications included temperature tracking, thermal regulation, melt-pool geometry control, and spatial height control.Source: §I · p. 2

II. PROBLEM FORMULATION

The formulation treats dual-laser LPBF as layer-to-layer control of a single straight scan line, with the trailing laser reheating material during solidification. It optimizes four process inputs to track a target cooling profile at sampled control points, using spatial invariance to represent the track with an interior cell.

II. PROBLEM FORMULATION

  • The study focuses on layer-to-layer control on a single straight-line scan, a building block for hatching patterns; parameters are updated between trials rather than during a scan.The authors note that within-scan control would require high-bandwidth thermal monitoring, while layer-to-layer control can use existing machines.Source: §II · p. 2
  • The four inputs are leading and trailing laser powers, lag distance, and scan velocity; powers set energy input, lag sets reheat timing, and velocity sets interaction time.The stated ranges are 25–100 W for each power, 0–250 µm for lag distance, and 0.25–1.0 m/s for scan velocity.Source: §II · p. 2
  • The output is temperature at control points during cooling at a centreline cell, anchored to the last downward crossing of the liquidus to capture final solidification despite possible remelting.Outside a small boundary region, spatial invariance makes cells’ responses essentially time-shifted copies, so optimizing one interior cell represents the track.Source: §II · p. 3
  • The plant provides thermal responses from experiments, while a simulation approximating the plant provides a Jacobian through automatic differentiation for optimization.This couples plant observations with model derivatives in the formulation.Source: §II · p. 3
  • The objective tracks six temperature points through the solidification range, targeting a representative cooling rate of −2.5 MK/s from the 1600 K liquidus.The solidus is 1290 K; the points are sampled at offsets of 0, 40, 80, 120, 160, and 240 µs after the profile’s liquidus crossing, with target temperatures from 1600 K to 1100 K.Source: §II · p. 3

III. ITERATIVE LEARNING CONTROLLER

The controller minimizes thermal-profile tracking error by combining model-based search directions with plant feedback that corrects for model mismatch. It updates process parameters between repeated scans using a constrained optimization procedure.

III. ITERATIVE LEARNING CONTROLLER

  • The objective is to minimize tracking error between plant temperatures at control points and their target temperatures through the solidification region.The targets are specified at m control points.Source: §III · p. 3
  • Each SQP step uses the model Jacobian in a constrained QP, with the plant-model output residual correcting the linearized constraint.Eliminating the predicted-output increment reduces the QP to the n design variables.Source: §III · pp. 3–4
  • Each iteration requires one plant evaluation and one model Jacobian computation, and the constraint set keeps the updated design within U.The Jacobian is supplied by the differentiable simulation.Source: §III · pp. 3–4
  • The reduced gradient combines the model Jacobian with the plant residual, so plant feedback enters the search direction at every iteration.The reduced formulation avoids forming the full-space Hessian and requires an n×n BFGS update.Source: §III · p. 4
  • The reduced Hessian is updated by BFGS using design-step and gradient changes, with Powell damping chosen to preserve positive definiteness.The reduced QP is solved for the parameter step before the Hessian update.Source: §III · p. 4
  • The model-only solution provides a warm start, after which plant measurements guide parameter updates between repeated manufacturing trials.Each iteration corresponds to a scan, and the compensator is closed-loop with the plant.Source: §III · p. 4

IV. SIMULATION STUDY

The simulation study uses a high-fidelity thermal model as the plant and a deliberately mismatched low-fidelity model to supply optimization gradients. It compares model-only optimization with iterative learning control using plant feedback.

IV. SIMULATION STUDY

  • The high-fidelity simulation serves as the plant, while the low-fidelity simulation supplies model Jacobians and includes deliberate physical mismatches.Seven random starting points are used to characterize convergence reliability.Source: §IV · p. 4
  • The plant and model differ in absorptivity, latent-heat treatment, and effective thermal conductivity, with cost and reported metrics evaluated from 1290 to 1600 K.The plant uses η = 0.25, latent heat, and k = 8 W/(m·K); the model uses η = 0.20, no latent heat, and k = 15 W/(m·K).Source: §IV · p. 5
  • Low-fidelity model parameters are not guaranteed to be optimal or even good on the plant when applied without correcting for model mismatch.The mismatched thermal response is illustrated by the plant-model comparison in Figure 5.Source: §IV · p. 5, Figure 5 · p. 5
  • At identical input, the plant has a reduced peak temperature and a structurally different cooling rate through the solidification region.The comparison is between plant HF and model LF responses.Source: Figure 5 · p. 5
  • Model-only optimization applies its low-fidelity result to the plant without feedback, whereas ILC uses low-fidelity gradients with iterative plant feedback.The model-only case represents offline simulation tuning followed by open-loop application.Source: §IV · p. 5

V. SINGLE-LINE GEOMETRY: RESULTS

The study evaluates the controller on a powder-bed mismatch configuration using seven initial conditions for both ILC and model-only optimization.

V. SINGLE-LINE GEOMETRY: RESULTS

  • Seven initial conditions are used for each of ILC and model-only optimization in the powder-bed mismatch configuration.The configuration is described in Section IV.Source: §V · p. 5

A. Initial conditions (seeds) and convergence

Across seven seeds, ILC reduces plant cost by an order of magnitude and converges to a tight band, while model-only optimization’s plant reevaluation remains higher. The separation between model and plant costs reflects a mismatch that ILC corrects using plant feedback.

A. Initial conditions (seeds) and convergence

B. Convergence history at a single seed

At seed 7, ILC lowers plant cost by over an order of magnitude, while MO’s plant reevaluation remains near 10^5 K2 despite a modest decline on its low-fidelity objective. The methods converge to qualitatively different parameter regions.

B. Convergence history at a single seed

C. Thermal profile tracking

On the plant at seed 7, ILC tracks the target cooling profile more closely than either the single-laser baseline or MO. The methods differ because ILC evaluates its residual on the plant, while MO uses its model residual.

C. Thermal profile tracking

  • MO and ILC solve the same reduced QP but form their reduced gradients from different residuals: model residual for MO and plant residual for ILC.The low-fidelity model loses heat faster than the plant, so their residuals and resulting gradients point in opposite directions.Source: C. Thermal profile tracking · p. 6
  • Plant-evaluated residuals compensate for thermal-response prediction errors across iterations, provided model gradients remain descent directions for plant cost.Source: C. Thermal profile tracking · p. 6
  • ILC’s mean cooling rate is −2.34 MK/s against the −2.50 MK/s target, a 6% deviation.The plant-evaluated profile tracks the target control points through the solidification region.Source: Figure 8 · p. 6, C. Thermal profile tracking · p. 6
  • The single-laser baseline averages −2.96 MK/s, 18% faster than target, while MO averages −1.36 MK/s and remains well above the target profile.MO cools at roughly half the target rate.Source: Figure 8 · p. 6, C. Thermal profile tracking · p. 6

VI. NON-SINGLE-LINE GEOMETRY

The single-line ILC settings were transferred without re-optimization to a three-stroke zigzag scan and tracked the target cooling profile more closely than the single-laser baseline at both monitors. The vertex was the more difficult transfer location, where heat accumulates at the scan reversal.

VI. NON-SINGLE-LINE GEOMETRY

  • The three-stroke zigzag test holds parameters optimized on a single straight line fixed and applies them without re-optimization.The turn vertex accumulates heat from scan-direction reversal.Source: §VI · pp. 6–7
  • At the stroke midpoint and turn vertex, the controlled scan extends time in the solidification region by 480% and 364%, respectively, relative to baseline.The vertex is the harder transfer case and runs hotter than the target, but remains closer to it than baseline.Source: §VI · p. 6, Figure 9 · p. 7
  • The controlled zigzag tracks the target cooling profile more closely than the too-fast-cooling single-laser baseline at both monitors.The monitors are the stroke midpoint and turn vertex; controlled peak temperatures are higher, but peak temperature is not the controlled quantity.Source: §VI · p. 6, Figure 9 · p. 7

VII. CONCLUSION

The SQP-based iterative learning controller tracks a cooling profile favoring equiaxed grains by optimizing dual-laser powers, lag distance, and scan velocity. Across seven initial conditions, it reaches a tight band of plant solutions despite realistic model mismatch, while model-only optimization overshoots the target cooling profile on the plant.

VII. CONCLUSION

  • CV 5.6% across 7 initial conditions: ILC reaches a tight band of plant solutions within roughly 20 iterations despite realistic model mismatch.The mismatch includes absorption, latent heat treatment, and powder-bed conductivity.Source: §VII · p. 7
  • Model-only optimization converges to the low-fidelity optimum, but its parameter selections overshoot the target cooling profile when evaluated on the plant.Source: §VII · p. 7
  • The framework tracks a cooling profile favoring equiaxed grains by optimizing leading and trailing laser powers, lag distance, and scan velocity.It requires no algorithmic changes for deployment on hardware with pyrometer feedback.Source: §VII · p. 7
  • Future work includes experimental validation and extension to spatially varying control along the scan path.Source: §VII · p. 7
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