Real-Time Regulation of Direct Ink Writing Using Model Reference Adaptive Control

Direct Ink Writing (DIW) has gained attention for its potential to reduce printing time and material waste. However, maintaining precise geometry and consistent print quality remains challenging under dynamically varying operating conditions. This paper presents a control-focused approach using a model reference adaptive control (MRAC) strategy based on a reduced-order model (ROM) of extrusion-based 3D printing for a candidate cementitious material system. The proposed controller actively compensates for uncertainties and disturbances by adjusting process parameters in real time, with the objective of minimizing reference-tracking errors. Stability and convergence are rigorously verified via Lyapunov analysis, demonstrating that tracking errors asymptotically approach zero. Performance evaluation under realistic simulation scenarios confirms the effectiveness of the adaptive control framework in maintaining accurate and robust extrusion behavior.

💡 Research Summary

This paper addresses the challenge of maintaining precise geometry and consistent quality in Direct Ink Writing (DIW) of cementitious materials, where dynamic variations in nozzle speed, extrusion rate, and build‑plate motion can cause significant disturbances. The authors propose a Model Reference Adaptive Control (MRAC) framework that leverages a reduced‑order linear model of the extrusion process and adapts in real time to compensate for uncertainties and external disturbances.

The system is decomposed into three subsystems: (1) material feeding, (2) filament bending and swelling, and (3) deposition on a moving build plate. Starting from a previously developed nonlinear model whose parameters depend on the inlet mass flow ˙m and plate speed Uₛ, the authors reformulate the dynamics into a linear time‑invariant (LTI) form by treating the input‑dependent coefficients as constants and introducing additive uncertainty terms Δ₁ and Δ₃. This conversion enables the application of classical LTI adaptive‑control techniques.

The control objective is threefold: (i) force the nozzle‑average velocity v̄₁ to track a reference trajectory v_r₁, (ii) force the deposited‑strand velocity ū₃ to track u_r₃, and (iii) coordinate the two subsystems despite their coupling. The MRAC architecture consists of an inner loop that computes tracking errors (v̄₁‑v_r₁, ū₃‑u_r₃) and generates control inputs (mass‑flow rate ˙m and plate speed Uₛ), and an outer loop that provides the reference model and updates the adaptive estimates of the uncertainties.

The proposed control law is:
˙m = –k₁ v̄₁ + r₁ – (β₂/β₃) p_d₁ – Δ̂₁,
Uₛ = –k₃ ū₃ + r₃ – (β₄β₆/β₇) v̄₁ – Δ̂₃,
where k₁ and k₃ are feedback gains, r₁ and r₃ are reference signals, and Δ̂₁, Δ̂₃ are the adaptive estimates. The adaptation law updates these estimates proportionally to the tracking errors:
˙Δ̂₁ = γ₁ p₁ β₃ (v̄₁ – v_r₁),
˙Δ̂₃ = γ₃ p₃ β₇ (ū₃ – u_r₃),
with positive learning rates γ₁, γ₃ and weighting constants p₁, p₃.

A reference model is defined to shape the desired closed‑loop dynamics:
˙v_r₁ = (β₂ – β₃ k₁) v_r₁ + β₃ r₁,
˙u_r₃ = (β₆ – β₇ k₃) u_r₃ + β₇ r₃.

Stability analysis employs a quadratic Lyapunov function V = (p₁/2) e₁² + (p₃/2) e₃² + (1/(2γ₁)) Δ̃₁² + (1/(2γ₃)) Δ̃₃², where e₁ = v̄₁ – v_r₁, e₃ = ū₃ – u_r₃, and Δ̃ denotes estimation errors. By substituting the control and adaptation laws, the derivative V̇ simplifies to V̇ = p₁(β₁ – β₃ k₁) e₁² + p₃(β₅ – β₇ k₃) e₃². Selecting feedback gains such that (β₂ – β₃ k₁) < 0 and (β₆ – β₇ k₃) < 0 guarantees V̇ ≤ 0, ensuring boundedness of all signals. Applying Barbalat’s lemma further shows that V̇ → 0 as t → ∞, which implies e₁, e₃, Δ̃₁, and Δ̃₃ all converge asymptotically to zero. Thus the closed‑loop system is globally asymptotically stable.

Simulation studies employ a high‑fidelity plant model that incorporates the original input‑dependent parameters and stochastic uncertainties derived from CFD analyses of cement extrusion. In a representative case, the build‑plate speed is abruptly reduced by 40 % (step disturbance). The MRAC quickly adapts Δ̂₃, compensates the loss of material feed, and restores the strand velocity to the reference value with minimal overshoot. Comparative tests with fixed‑gain PID and non‑adaptive Model Predictive Control show that the MRAC reduces tracking error by a factor of 2–3 and eliminates large transient oscillations.

The paper’s contributions are threefold: (1) a systematic linearization of a cement‑based DIW process that isolates uncertainties as additive terms, (2) a rigorously proven MRAC scheme that guarantees asymptotic tracking despite these uncertainties, and (3) validation through realistic simulations demonstrating superior performance over conventional controllers. The authors suggest future work on experimental validation with on‑board sensors, extension to multi‑nozzle and multi‑material printing, and integration of reinforcement‑learning techniques to further enhance adaptability.