Threshold dynamics in time-delay systems: polynomial $β$-control in a pressing process and connections to blow-up

TL;DR

The paper tackles a time-delay press-control problem in straightening machines and introduces a generalized polynomial $β$-control to replace linear velocity rules, resulting in a delay differential equation. Through nondimensionalization, the authors establish global existence and reveal a threshold $g(β)$ that separates overshoot from asymptotic convergence, supported by numerical experiments and a proposed conjecture. Under velocity constraints, they design a practical control algorithm using this threshold, providing a guiding estimate $v_0\approx(ℓ/τ)g(β)$ for tuning and demonstrating improved performance. Additionally, they connect the threshold dynamics to finite-time blow-up in DDEs, offering insights into blow-up rates and the interplay between delay, control, and stability, while outlining paths for extending the model to time-varying delays.

Abstract

This paper addresses a press control problem in straightening machines with small time delays due to system communication. To handle this, we propose a generalized $β$-control method, which replaces conventional linear velocity control with a polynomial of degree $β\ge 1$. The resulting model is a delay differential equation (DDE), for which we derive basic properties through nondimensionalization and analysis. Numerical experiments suggest the existence of a threshold initial velocity separating overshoot and non-overshoot dynamics, which we formulate as a conjecture. Based on this, we design a control algorithm under velocity constraints and confirm its effectiveness. We also highlight a connection between threshold behavior and finite-time blow-up in DDEs. This study provides a practical control strategy and contributes new insights into threshold dynamics and blow-up phenomena in delay systems.

Figures (6)

• Figure 1: The target pressing stroke (red line) and the actual sensor position (blue line), illustrating the overshoot phenomenon due to time delay. The press is stopped after overshooting.
• Figure 2: Schematic of signal flow in the straightening press machine.
• Figure 3: Example of press machine data. (a) Comparison of the actual and commanded press velocities. The actual velocity is valid only after $t = 250$ ms. (b) Estimation of time delay $\tau$ as the horizontal gap between the two curves. The average delay is approximately $\tau = 40$ ms.
• Figure 4: Numerical results with $\beta = 2$ and $\Delta t = 0.01$ (red, green, and blue lines), compared to the constant value 1 (black line). The red, green, and blue lines show the evolution of $u(t)$ over $t \in [0,10]$ for $w_0 = 0.55$, $0.66$, and $0.77$, respectively.
• Figure 5: Log-log plots of $1 - g(\beta)$ and its linear approximation over $\beta \in [1,100]$ (a) and $\beta \in [4,100]$ (b), and numerical simulations with $w_0 = g(\beta)$ for $\beta = 1,2,5,10,50$ (c).

Theorems & Definitions (12)

• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Conjecture 3.3
• Proposition 3.4
• proof
• Remark 4.3
• Theorem 5.1
• proof