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A Digital Twin of Evaporative Thermo-Fluidic Process in Fixation Unit of DoD Inkjet Printers

Samarth Toolhally, Joeri Roelofs, Siep Weiland, Amritam Das

TL;DR

Problem addressed: achieving optimal paper moisture and temperature control in the fixation unit of inkjet printers requires a faithful, real-time model of the spatio-temporal thermo-fluidic drying. Approach: model the unit as a modular graph with six spatial zones, convert diffusion and evaporation dynamics into a unified Partial Integral Equation framework, and synthesize an H-infinity optimal Luenberger estimator for robust state observation. Contributions: a validated PIE-based digital twin implemented in MATLAB via PIETOOLS, an H-infinity estimator with a quantified robustness bound, and a modular architecture that supports rapid prototyping. Significance: enables reliable real-time monitoring and potential control improvements for industrial inkjet fixation, while outlining current limitations and directions for extending to two-dimensional diffusion and more complete nonlinearities.

Abstract

In inkjet printing, optimal paper moisture is crucial for print quality, achieved through hot-air impingement in the fixation unit. This paper presents a modular digital twin of the fixation unit, modeling the thermo-fluidic drying process and monitoring its spatio-temporal performance. The novel approach formulates the digital twin as an infinite-dimensional state estimator that infers fixation states from limited sensor data, while remaining robust to disturbances. Modularity is achieved through a graph-theoretic model, where each node represents thermo-fluidic dynamics in different sections of the fixation unit. Evaporation is modeled as a nonlinear boundary effect coupled with node dynamics via Linear Fractional Representation. Using the Partial Integral Equation (PIE) framework, we develop a unified approach for stability, input-output analysis, simulation, and rapid prototyping, validated with operational data from a commercial printer. An $\mathcal{H}_{\infty}$-optimal Luenberger state estimator is then synthesized to estimate thermal states from available sensor data, enabling real-time monitoring of spatio-temporal thermal effects on paper sheets.

A Digital Twin of Evaporative Thermo-Fluidic Process in Fixation Unit of DoD Inkjet Printers

TL;DR

Problem addressed: achieving optimal paper moisture and temperature control in the fixation unit of inkjet printers requires a faithful, real-time model of the spatio-temporal thermo-fluidic drying. Approach: model the unit as a modular graph with six spatial zones, convert diffusion and evaporation dynamics into a unified Partial Integral Equation framework, and synthesize an H-infinity optimal Luenberger estimator for robust state observation. Contributions: a validated PIE-based digital twin implemented in MATLAB via PIETOOLS, an H-infinity estimator with a quantified robustness bound, and a modular architecture that supports rapid prototyping. Significance: enables reliable real-time monitoring and potential control improvements for industrial inkjet fixation, while outlining current limitations and directions for extending to two-dimensional diffusion and more complete nonlinearities.

Abstract

In inkjet printing, optimal paper moisture is crucial for print quality, achieved through hot-air impingement in the fixation unit. This paper presents a modular digital twin of the fixation unit, modeling the thermo-fluidic drying process and monitoring its spatio-temporal performance. The novel approach formulates the digital twin as an infinite-dimensional state estimator that infers fixation states from limited sensor data, while remaining robust to disturbances. Modularity is achieved through a graph-theoretic model, where each node represents thermo-fluidic dynamics in different sections of the fixation unit. Evaporation is modeled as a nonlinear boundary effect coupled with node dynamics via Linear Fractional Representation. Using the Partial Integral Equation (PIE) framework, we develop a unified approach for stability, input-output analysis, simulation, and rapid prototyping, validated with operational data from a commercial printer. An -optimal Luenberger state estimator is then synthesized to estimate thermal states from available sensor data, enabling real-time monitoring of spatio-temporal thermal effects on paper sheets.

Paper Structure

This paper contains 13 sections, 2 theorems, 20 equations, 11 figures, 4 tables.

Table of Contents

  1. Preliminaries
  2. Modeling the Thermo-Fluidic Process in Fixation
  3. Graph theoretic representation
  4. Node Structure:
  5. Edge Structure:
  6. Graph-Theoretic Model of the Fixation Unit
  7. Implementation and simulation of the model
  8. Comparison of the fixation process model with machine data
  9. Simulation results for $\mathrm{g/m^2}$ blank paper:
  10. Simulation results for 115 $\mathrm{g/m^2}$ printed paper:
  11. Digital Twining of Fixation Process through $\mathcal{H}_{\infty}$ Optimal State Estimator Synthesis
  12. $\mathcal{H}_{\infty}$ optimal estimator for blank paper
  13. Conclusions

Key Result

Proposition 2

The thermo-fluidic model derived using Definition def1, described by the dynamic equations dynamicmain, and boundary conditions bctemp and bcmoist is equivalent to a dynamical system $\mathfrak{P}_{\mathrm{p}}$ whose behavior is governed by Partial Integral Equations (PIE) as follows: Here, $\mathscr{T}, \mathscr{T}_w, \mathscr{T}_d$, $\mathscr{A}, \mathscr{B}_{11}, \mathscr{B}_{12}, \mathscr{B}

Figures (11)

  • Figure 2: The graph theoretical representation of the paper-conveyor system. ,, represent the paper, conveyor and the air layers respectively. $w_1(t)$ is the unknown disturbance, $d_1(t),d_2(t)$ are the known disturbance acting at the boundaries. $\Delta$ is the non-linearity of the system.
  • Figure 3: Equivalence between the developed model and PIEs
  • Figure 4: Schematic of the LFR structure. $\Delta_1(p(t))$ is a static non-linear function that computes the moisture flux evaporating, and $\Delta_2(p(t))$ is a static non-linearity that computes the enthalpy required to change the state of the moisture to its vapour.
  • Figure 5: Temperature variation over space and time for all the nodes of the 350 $\mathrm{g/m^2}$ paper and conveyor. () represents the boundary for each node. The black arrow shows the node with ink and moisture.
  • Figure 6: Comparison of simulation results and sensor readings for 350 $\mathrm{g/m^2}$ paper.
  • ...and 6 more figures

Theorems & Definitions (6)

  • Definition 1
  • Proposition 2
  • Proof 3.1
  • Proposition 3
  • Proof 4.1
  • Remark 4