# Robust Additive Manufacturing Performance through a Control Oriented Digital Twin

^{*}

## Abstract

**:**

## 1. Introduction

#### State of the Art on Manufacturing Processes Monitoring and Control

## 2. Approach

#### 2.1. Simulation

#### 2.2. Dynamic System Identification of Simulation

#### 2.3. Process Control of Laser-Based Manufacturing Processes

#### 2.4. Uncertainty Manipulation through Robustness

- Disturbances existence;
- Noise existence;
- Uncertainties existence;
- Delays existence.

- H-Infinity Optimal Dynamic Output Feedback Control;
- H-2 Optimal Dynamic Output Feedback Control;
- Observed-based controller (H-Infinity state-feedback gain ${K}_{\mathit{obs}}$ and observer; gain ${L}_{\mathit{obs}}$) in the form as described by Apkarian and Noll [55];
- Robust MPC with polytopic uncertainties.

## 3. Case Study: Robust Digital Twin for LPBF

#### 3.1. Heat Transfer of the Model

^{2}), where ${r}_{o}$ is the beam’s radius, $a$ is the absorbance derived from ray-tracing in powder, ${P}_{\mathit{laser}}$ is the power of laser, and $r$ is the radial distance from the laser beam center and is described by Equation (4).

#### 3.2. Boundary and Initial Conditions

#### 3.3. Upper Free Side

^{2}K [61], and the coefficient of radiation emissivity is assumed to be temperature-dependent [62].

#### 3.4. Left and Right Side

#### 3.5. Lower Side

#### 3.6. Systemic Modelling and Design of Controllers

_{p}: (A, B, C, D).

_{uncert}≤ 10%. To further elaborate on this, five discrete absolute values were selected in order to reduce the computational time of FEM and the RMPC algorithm, namely −10%, −7.5%, −5%, −2.5%, 0% (nominal process), 2.5%, 5%, 7.5%, and 10%. Each simulation provides a response of the maximum temperature on the melt-pool and the relationship between input-output, as described by an ARX model. Hence, eight different mathematical models occur in the uncertain process description and a single one as the nominal one. Then, we considered the following linear parameter-varying (LPV) state-space in order to synthesize the Robust MPC algorithm under the assumption of known uncertainties from Hu and Ding [54], as shown in the Equation (11), and the controller has the form of the Equation (10).

## 4. Results and Discussion

#### 4.1. System Representation

#### 4.2. Performance of the Initial Digital Twin

_{2}, and the observer-based controllers produced similar dynamic power signals. The assessment of all the controllers is conducted utilizing typical metrics of the step response, as depicted in Figure 11a, while Figure 11b presents the comprehensive results from the all the developed controllers and the empirical power signal. The comprehensive results of the initial Digital Twin are presented in the Table 2. To illustrate the contrast in dynamical profile of power with the constant one, the step response metrics are investigated.

#### 4.3. Robustness of the LPBF DT

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

Acronym | Definition |

AM | Additive Manufacturing |

ARX | AutoRegressive with eXogenous (parametric system identification method) |

CCD | Charge-coupled device |

CMOS | Complementary metal–oxide–semiconductor |

DOF | Dynamic Output Feedback controller |

DT | Digital Twin |

FEM | Finite Element Method |

H_{2} | Optimal H_{2} Controller |

H_{∞} | Optimal H_{∞} Controller |

KPI | Key Performance Indicator |

LMI | Linear Matrix Inequality |

LPBF | Laser-based Powder Bed Fusion |

PID | Proportional-Integral-Differential controller |

RMPC | Robust Model Predictive Control |

SISO | Single-Input-Single-Output control system |

## Appendix A

**Table A1.**Material Properties of Ti-6Al-4V–Adopted from Mills (2002) [70].

Property | Unit | Nomenclature | Value |
---|---|---|---|

Liquidus Temperature | K | Tl | 1923.0 |

Solidus Temperature | K | Ts | 1877.0 |

Evaporation Temperature | K | Tv | 3533.0 |

Solid Specific Heat | J/kgK | Cs | $\{\begin{array}{l}483.04+0.22T\hspace{1em}T\le 1268K\\ 412.70+0.18T\hspace{1em}1268KT1923K\end{array}$ |

Liquidus Specific Heat | J/kgK | Cl | 831.0 |

Thermal Conductivity | W/mK | K | $\{\begin{array}{l}1.260+0.016T\hspace{1em}\hspace{1em}T\le 1268K\\ 3.513+0.013T\hspace{1em}\hspace{1em}1268KT1923K\\ -12.752+0.024T\hspace{1em}T\ge 1923K\end{array}$ |

Solid Density | kg/m^{3} | ρ_{s} | 4420-0.154 (T-298 K) |

Liquidus Density | kg/m^{3} | ρ_{l} | 3920-0.68 (T-1923 K) |

Latent heat of fusion | J/kg | Lm | 2.86 × 10^{5} |

Radiation emissivity | – | ε | 0.154 + 1.838 × 10^{-4}(T-300 K) |

Absorption | – | α | 0.3 |

Ambient Temperature | K | T0 | 293.15 |

Ambient Pressure | Pa | P0 | 1.0 × 105 |

Convective coefficient | W/m^{2}K | H | 10.0 |

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**Figure 1.**Evolution of publications number regarding additive manufacturing (AM), based on Scopus database and query (TITLE-ABS-KEY(additive manufacturing)). Accessed on 18 March 2021.

**Figure 2.**Evolution of publications regarding AM and control, based on Scopus database and query: (TITLE-ABS-KEY ("additive manufacturing" and "control")). Accessed on 18 March 2021.

**Figure 6.**Schematic of LPBF System (own illustration, close-up of powder can be found in [50]).

**Figure 9.**Estimated systemic models. The solid line indicates the initial measurement from FEM, dashed line indicates the estimated models (

**a**), close-up of systemic models (

**b**).

**Figure 11.**Step response metrics (

**a**), robust optimization solution based on LMIs in comparison with a fine-tuned PID controller (

**b**).

Process Parameters | Units | Nomenclature | Value |
---|---|---|---|

Laser beam radius | μm | R_{o} | 50 |

Laser power | W | P_{o} | 50 |

Laser speed | mm/s | V | 750 |

Laser power peak | μs | Δt_{on} | 100 |

Laser power duration | μs | Δt | 1000 |

Reference | Controller | Settling Time (μs) | Rise Time (μs) | Overshoot (%) | Comparison to Settling Time (%) | Comparison to Rise Time (%) |
---|---|---|---|---|---|---|

Current work | Initial | 140 | 82.1 | 0.9 | – | – |

H-Infinity | 45.2 | 29.4 | 1.1 | 67.7% | 64.2% | |

H2 | 45.3 | 29.5 | 1.4 | 67.6% | 64.1% | |

Observer-based H-Infinity | 45.1 | 29.4 | 0.9 | 67.8% | 64.2% | |

PID | 86.7 | 55.6 | 1.5 | 38.1% | 32.3% | |

[65] | Constant parameters | ~250.0 | ~125.0 | – | – | – |

[66] at the case of 200 W, 800 mm/s | ~150.0 | ~75.0 | – | – | – |

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**MDPI and ACS Style**

Stavropoulos, P.; Papacharalampopoulos, A.; Michail, C.K.; Chryssolouris, G. Robust Additive Manufacturing Performance through a Control Oriented Digital Twin. *Metals* **2021**, *11*, 708.
https://doi.org/10.3390/met11050708

**AMA Style**

Stavropoulos P, Papacharalampopoulos A, Michail CK, Chryssolouris G. Robust Additive Manufacturing Performance through a Control Oriented Digital Twin. *Metals*. 2021; 11(5):708.
https://doi.org/10.3390/met11050708

**Chicago/Turabian Style**

Stavropoulos, Panagiotis, Alexios Papacharalampopoulos, Christos K. Michail, and George Chryssolouris. 2021. "Robust Additive Manufacturing Performance through a Control Oriented Digital Twin" *Metals* 11, no. 5: 708.
https://doi.org/10.3390/met11050708