[Paper Review] Event-Triggered Control of a Continuum Model of Highly Re-Entrant Manufacturing System

Mamadou Diagne, Iasson Karafyllis|arXiv (Cornell University)|Apr 4, 2020

Stability and Control of Uncertain Systems45 references4 citations

TL;DR

This paper proposes an event-triggered boundary controller for a continuum model of highly re-entrant manufacturing systems using a Lyapunov-based design to stabilize product density to a uniform equilibrium. The controller reduces communication and control updates by triggering only when necessary, ensuring global stability and avoiding Zeno behavior, with robustness verified via sampled-data simulations and convergence to equilibrium in the L² norm.

ABSTRACT

With the unceasing growth of intelligent production lines that integrate sensors, actuators, and controllers in a wireless communication environment via internet of things (IoT), we design an event-triggered boundary controller for a continuum model of highly re-entrant manufacturing systems for which the influx rate of products is the controlled quantity. The designed controller can potentially operate in networked control systems subject to limited information sharing resources. A Lyapunov argument is utilized to derive the boundary controller together with a feasible event generator that avoids the occurrence of Zeno behavior for the closed-loop system. The global stability estimate is established using the logarithmic norm of the state due to the system's nonlinearity and positivity of the density. Furthermore, robustness of the proposed controller with respect to the sampling schedule and sampled-data stabilization results are established. Consistent simulation results that support the proposed theoretical statements are provided.

Motivation & Objective

• To address the challenge of stabilizing highly re-entrant manufacturing systems with high work-in-progress fluctuations using a continuum model.
• To design an event-triggered boundary controller that minimizes control updates under limited communication and computational resources.
• To ensure global stability and avoid Zeno behavior in the closed-loop system using a Lyapunov argument.
• To establish robustness of the controller with respect to sampling schedule and sampled-data implementation.
• To validate theoretical results through consistent simulations of density dynamics and inter-execution times.

Proposed method

• A Lyapunov function is constructed to derive a boundary controller that stabilizes the system to a uniform equilibrium density.
• An event generator is designed based on a Lyapunov-based triggering condition to determine when control updates are necessary.
• The nonlocal PDE model describes product density evolution with a propagation speed dependent on total work-in-progress, defined as λ(W) = 1/(1+W).
• The controller uses the influx rate at the boundary x=0 as the control input, with the state governed by a nonlocal hyperbolic PDE.
• Robustness is analyzed by applying the same controller with periodic sampling, simulating a sampled-data implementation.
• Simulations use initial density ρ₀(x) = 6 + sin(πx) and equilibrium ρₛ = 1, with σ as a tuning parameter for event frequency.

Experimental results

Research questions

• RQ1 Can an event-triggered control strategy stabilize a continuum model of a highly re-entrant manufacturing system with nonlocal PDE dynamics?
• RQ2 How can a Lyapunov-based event generator ensure Zeno-free operation while maintaining global stability?
• RQ3 What is the impact of the event threshold σ on convergence speed and control update frequency?
• RQ4 How robust is the event-triggered controller to variations in the sampling schedule?
• RQ5 Can the same controller be effectively implemented in a sampled-data setting with periodic updates?

Key findings

• The event-triggered controller successfully stabilizes the product density to the uniform equilibrium ρₛ = 1 for both σ = 0.02 and σ = 0.006, as confirmed by L² norm convergence.
• Higher σ values (e.g., 0.02) lead to faster convergence due to increased control update frequency, as shown in Figure 5.
• The inter-execution time statistics indicate that for σ = 0.02, most intervals fall in [0.6, 1], while for σ = 0.006, they are in [0.6, 2], suggesting conservative sampling period choices.
• Sampled-data simulations with T = 1 and T = 2.5 confirm robustness, showing stable convergence of input density and output flux to equilibrium.
• The L² norm of the state deviation tends to zero over time, confirming L² stability under both event-triggered and sampled-data control.
• The distributed density dynamics in Figures 6 and 7 show that ρ(t,x) converges uniformly to ρₛ = 1 across the spatial domain for both σ values.