Multi-layer predictor feedback design for nonlinear integro-differential equations with state-dependent input delays

Abstract

We develop a novel multi-layer predictor feedback to achieve exact compensation of state-dependent input delays 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. We first address a source term corresponding to a linear recirculation loop. The framework is subsequently generalized to source terms such as a nonlinear recirculation loop and a linear friction term. In addition, in the latter case, we use boundary flux control. We show that the inhomogeneous and doubly nonlinear mixed PDE-ODE system results in a correspondingly “inhomogeneous” control law. The stabilizing controller comprises a nominal delay-free control law based on the predictor feedback state, enhanced by a nonlinear multi-layer prediction mechanism that compensates for the PDE’s additive source terms. Our modular designs are based on two novel nonlinear backstepping transformations: the first enables boundary control of the PDE state, while the second addresses flux control—namely, a nonlinear controlled boundary condition defined as the product of the ODE and PDE states. 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.