⚡ electrical engineering

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

This paper presents a modular, graph-theoretic digital twin for the fixation unit of DoD inkjet printers that utilizes a Partial Integral Equation framework and an $\mathcal{H}_{\infty}$ -optimal state estimator to robustly infer spatio-temporal thermal states from limited sensor data for real-time monitoring of the evaporative thermo-fluidic drying process.

Original authors: Samarth Toolhally, Joeri Roelofs, Siep Weiland, Amritam Das

Published 2026-03-26

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Samarth Toolhally, Joeri Roelofs, Siep Weiland, Amritam Das

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: The "Magic Oven" Problem

Imagine a high-speed printing press as a giant, high-tech bakery. In this bakery, a conveyor belt carries sheets of paper (the "dough") through a hot oven (the "fixation unit").

The Goal: The printer sprays ink onto the paper. To make the ink stick and look sharp, the paper needs to be dried perfectly. Too wet? The ink smears. Too dry? The paper curls or cracks.
The Problem: The oven is moving so fast, and the paper is so thin, that you can't stick a thermometer inside the paper while it's baking. You can only measure the temperature of the air blowing on it and the temperature of the conveyor belt underneath.
The Consequence: Because the engineers can't "see" inside the paper, they are flying blind. They don't know if the center of the paper is soggy or if the top is burning. This leads to bad print quality.

The Solution: Building a "Digital Twin"

The authors (engineers from a university and Canon) decided to build a Digital Twin.

Think of a Digital Twin not as a robot, but as a hyper-realistic video game simulation of the real oven.

The Virtual Replica: They created a computer model that mimics exactly how heat and moisture move through the paper layers.
The "Ghost" Sensors: Since they can't put sensors inside the real paper, they use the Digital Twin to act as a "ghost sensor." The twin takes the limited data it can see (air temp, belt temp) and uses math to guess what is happening inside the paper.

How They Built the Model: The "Lego" Approach

Instead of trying to solve one giant, impossible math equation for the whole oven, they broke the problem down into Lego blocks.

The Graph Theory: They imagined the oven as a network of connected nodes (blocks).
- Block 1: The hot air above the paper.
- Block 2: The top layer of the paper.
- Block 3: The bottom layer of the paper.
- Block 4: The conveyor belt.
The Connection: They used a "graph" (a map of connections) to show how heat flows from the air to the paper, and from the paper to the belt. This makes the model modular. If the printer changes the type of paper (like switching from thin newsprint to thick cardboard), they just swap out the "Lego block" for the paper without rebuilding the whole engine.

The Secret Sauce: PIEs (The "Magic Translator")

This is the most technical part, but here is the simple version:

Usually, modeling heat and moisture involves Partial Differential Equations (PDEs). These are notoriously difficult to solve on a computer because they are "infinite-dimensional" (there are infinite points inside the paper). Traditional methods try to chop the paper into tiny chunks (like pixels) to solve it, but this is slow and often inaccurate.

The authors used a new mathematical tool called Partial Integral Equations (PIEs).

The Analogy: Imagine trying to describe a smooth, flowing river.
- Old Way (PDEs): You try to count every single water molecule. It takes forever and you might miss some.
- New Way (PIEs): You describe the flow of the river as a whole. You don't need to count molecules; you just describe the current.
Why it matters: This method allows the computer to solve the problem perfectly and quickly without chopping the paper into tiny, inaccurate pieces. It's like having a translator that turns a complex, messy language into a clean, simple one that computers love.

The "Sherlock Holmes" Estimator

Even with a perfect simulation, the model might drift away from reality if the wind changes or the heater flickers. So, they added a State Estimator.

The Metaphor: Think of this as Sherlock Holmes.
- Holmes (the estimator) has a theory about what's happening inside the paper (the Digital Twin).
- He gets clues from the limited sensors (the air temp, the belt temp).
- He compares his theory to the clues. If his theory says "It should be 80°C" but the sensor says "It's 78°C," Holmes adjusts his theory.
- The H∞ Optimization: This is the "Sherlock" being trained to be unshakeable. The engineers taught the estimator to handle the worst-case scenarios (like a sudden draft of cold air) without panicking. It guarantees that even if the world is chaotic, the guess about the paper's internal temperature will stay within a safe, accurate range.

The Results: Why This Matters

The team tested their "Ghost Sensor" on real Canon printers.

Accuracy: They compared the Digital Twin's guess of the paper's internal temperature against actual sensor readings. The guess was incredibly close (within 2°C).
Moisture Vision: Because they could accurately guess the temperature, they could also accurately guess the moisture inside the paper, which is impossible to measure directly.
The Future: This means printers can now "see" inside the paper in real-time. They can adjust the heat instantly to prevent curling or smearing, leading to higher quality prints, less waste, and faster speeds.

Summary in One Sentence

The authors built a smart, modular computer simulation that acts like a "X-ray vision" for industrial printers, using advanced math to guess the invisible temperature and moisture inside moving paper, ensuring perfect print quality without needing to touch the paper.

1. Problem Statement

In Drop-on-Demand (DoD) inkjet printing, the quality of the final print is heavily dependent on the fixation process, where printed paper is dried to achieve an optimal temperature-moisture balance. This process involves complex thermo-fluidic interactions (heat and moisture diffusion, evaporation) within the paper layers and the conveyor system.

Key Challenges:

Unobservable States: Direct measurement of the internal temperature and moisture profiles of the paper during high-speed fixation is impossible. Sensors can only measure surface temperatures (conveyor and top paper surface), leaving the internal "through-thickness" state unobservable.
Infinite-Dimensional Complexity: The physical phenomena are governed by Partial Differential Equations (PDEs). Traditional control and estimation methods often rely on "lumping" (discretizing the PDEs into ODEs), which leads to inaccurate robustness assessments and stability certificates, making them unreliable for industrial implementation.
Nonlinearity: Evaporation introduces strong nonlinear boundary effects that couple temperature and moisture dynamics, complicating modeling and control.
Lack of Real-Time Monitoring: Current industry practices rely on design improvements rather than real-time system-theoretic monitoring and compensation for these effects.

2. Methodology

The authors propose a modular, physics-based Digital Twin that combines a graph-theoretic physical model with advanced state estimation techniques.

A. Graph-Theoretic Modeling

The fixation unit is modeled as a finite graph $G = (N, A, E)$ :

Nodes ( $N$ ): Represent physical subsystems (paper layers, conveyor belt, air layers). Each node is governed by its own PDEs (diffusion equations for heat and moisture).
Edges ( $E$ ): Represent interconnections and boundary conditions between nodes (e.g., thermal contact between paper and conveyor).
Modularity: This structure allows for scalable adaptation to different paper types (composite layers) and machine configurations.
Nonlinearity: Evaporation is modeled as a nonlinear boundary effect using a Linear Fractional Representation (LFR), where latent heat and mass flux are coupled to the state variables via static nonlinear functions ( $\Delta$ ).

B. Partial Integral Equation (PIE) Framework

To overcome the computational limitations of standard PDE solvers, the authors convert the PDE model into Partial Integral Equations (PIEs):

Advantages: PIEs eliminate the need for explicit boundary conditions and smoothness constraints in the formulation. They allow the use of Linear Matrix Inequalities (LMIs) for stability analysis and controller/estimator synthesis, similar to standard ODE systems but without approximation errors associated with lumping.
Implementation: The model is implemented using the open-source PIETOOLS toolbox in MATLAB, utilizing PI operators for algebraic manipulation.

C. $H_\infty$ Optimal State Estimation

To reconstruct the unobservable internal states (temperature and moisture profiles) from limited sensor data:

An $H_\infty$ -optimal Luenberger state estimator is synthesized.
Objective: Minimize the $L_2$ gain ( $\gamma$ ) between external disturbances (unknown variations in hot air) and the estimation error.
Robustness: The estimator is designed to be robust against unknown exogenous perturbations and initial condition mismatches.
Virtual Sensing: By estimating the internal temperature field, the digital twin can infer the moisture profile and evaporation rates, effectively acting as a virtual sensor.

3. Key Contributions

First Industrial PIE Implementation: This work provides the first evidence of the PIE framework being successfully implemented in an industrial setting (Canon Production Printing) for a commercial inkjet printer, moving beyond theoretical proofs of concept.
Modular Graph-Theoretic Digital Twin: A novel representation of the thermo-fluidic process that cleanly encodes heterogeneous material layers and interconnections, allowing for rapid prototyping and scalability.
Robust Infinite-Dimensional Estimation: The synthesis of an $H_\infty$ state estimator for a distributed parameter system (PDE/PIE) that guarantees robustness against disturbances without relying on inaccurate lumped approximations.
Validation with Real Data: The model and estimator are validated against operational data from a commercial printer, demonstrating high accuracy in predicting temperature dynamics.

4. Results

The proposed digital twin was validated using two test cases:

Case 1: Unprinted Paper (350 g/m²): A linear heat diffusion scenario.
Case 2: Printed Paper (115 g/m² with ink): Included nonlinear moisture transport and evaporation.

Performance Metrics:

Simulation Accuracy: The model achieved low Root Mean Square Errors (RMSE) when compared to sensor data:
- Paper Surface Temp: ~1.91°C (350 g/m²) and ~2.08°C (115 g/m²).
- Conveyor Temp: ~0.58°C (350 g/m²) and ~2.65°C (115 g/m²).
- Normalized RMSE (NRMSE) was consistently below 5%.
Estimator Performance:
- The $H_\infty$ gain ( $\gamma$ ) was calculated as 0.08, indicating strong disturbance rejection.
- The estimator successfully reconstructed the internal temperature profile (specifically the bottom surface of the paper, $z_3(t)$ ) even with unknown initial conditions.
- Maximum estimation errors were observed to be within acceptable ranges (e.g., ~0.35°C for matched initial conditions, increasing to ~49°C for zero initial conditions, but converging rapidly).

5. Significance and Impact

Quality Improvement: By enabling real-time monitoring of spatio-temporal thermal effects, the digital twin allows for better control of the drying process, directly improving print quality and reducing defects related to moisture and thermal stress.
Cost and Time Efficiency: The digital twin reduces the need for costly and time-consuming physical experiments for tuning fixation parameters across different paper types.
Theoretical Advancement: It bridges the gap between advanced control theory (PIE frameworks, infinite-dimensional estimation) and industrial application, proving that complex distributed parameter systems can be controlled and monitored robustly in real-time.
Future Potential: The framework is extensible to 2D diffusion and can incorporate other nonlinearities (like condensation), offering a pathway to optimize next-generation sustainable printing processes.

Conclusion:
This paper successfully demonstrates a modular, physics-based digital twin for DoD inkjet fixation. By leveraging the PIE framework for modeling and $H_\infty$ synthesis for estimation, the authors have created a system capable of inferring unobservable internal states with high accuracy and robustness, addressing a critical bottleneck in modern industrial printing.