Multi-layer Predictor Feedback Design for Nonlinear Integro-Differential Equations with State-dependent Input Delays

TL;DR

This paper introduces a multi-layer predictor-feedback control method for nonlinear integro-differential equations with state-dependent delays, ensuring global stability in complex PDE-ODE systems.

Contribution

It presents a novel predictor-feedback design that handles mixed PDE-ODE systems with state-dependent delays, using characteristic methods and fixed-point analysis.

Findings

Achieves global asymptotic stability in PDE and ODE states.

Effectively stabilizes buffer and queue systems with state-dependent production rates.

Demonstrates control effectiveness through numerical simulations of production and queuing systems.

Abstract

We develop a novel multi-layer predictor-feedback to achieve exact compensation of state-dependent input delay of general nonlinear integro-differential equations. The system of interest is an unconventional mixed Partial Differential Equation (PDE)-Ordinary Differential Equation (ODE) system, in which a nonlinear ODE is actuated through an inhomogeneous advection PDE. Moreover, the propagation speed of the PDE depends on a moving window integral of the ODE state. The two above features are not addressed yet in standard PDE backstepping-based predictor-feedback designs. Unlike the conventional Lyapunov-based approaches used in the field, our stability and well-posedness analysis rely on the characteristic method and a fixed-point argument. Both of our designs achieve global asymptotic stability (GAS) in the supremum norm of the PDE and ODE states under the mild assumption that the nonlinearity in the PDE governing equation is uniformly Lipschitz continuous. The transport speed, governed by the integral of the ODE state, models systems such as production or queuing processes in which the state of a finite buffer-namely, the inventory level-determines the production or service rate. Numerical simulations demonstrate the effectiveness of the proposed control design for buffer-regulated production lines and queuing systems, ensuring asymptotic stability under a locally safe softened bang-bang feedback law that preserves the positivity of both the system state and the actuation signal.