Reduced-Order Backstepping Control of Nonlinear Mechanical Systems with Input Delay via Galerkin Projection

Abstract

This paper addresses trajectory tracking for a class of nonlinear mechanical systems with input delay. The delayed actuator channel is represented as an equivalent transport partial differential equation, thereby transforming the original delayed system into a coupled plant-channel representation. A finite-dimensional approximation of the delay channel is then constructed through Galerkin projection, resulting in an explicit ODE model of the delayed actuator dynamics. Using this reduced-order plant-channel model, a backstepping controller is developed, and Lyapunov analysis establishes uniform exponential stability of the nominal tracking error dynamics for the reduced-order Galerkin model. To enhance practical implementation, a command-filtered version is also introduced to avoid direct differentiation of virtual controls, for which a corresponding uniform ultimate boundedness result is obtained. Simulation studies are provided to validate the proposed design and illustrate the effectiveness of the method.

arXiv commentAccepted for presentation at the 65th IEEE Conference on Decision and Control (CDC 2026)