On the Negative Imaginary Time-Delay Systems With Its Application to Flexible Systems Control

Abstract

A new lemma is proposed to determine the negative imaginary (NI) properties of time-delay systems of retarded and neutral types. The lemma is employed for stability analysis in the context of positive feedback interconnection of infinite-mode NI systems and time-delay systems. A cantilever beam with a collocated piezoelectric sensor and actuator pair is studied as an infinite-mode NI system. A novel time-delay controller is proposed to be used within positive position feedback (PPF) and parametrized based on H∞-norm optimization targeting the vibration suppression of the flexible beam. The proposed lemma demonstrates that the designed controller is NI. We also present a corollary to ensure the stability of the closed-loop beam system with the proposed controller using Lyapunov and dissipation based stability approaches. The theoretical results are further verified by simulations. Moreover, the efficacy of the controller is evaluated against a state-of-the-art controller for vibration suppression. Simulation results show that the proposed controller achieves better performance in suppressing vibrations without significant change in robustness. © 2025 Elsevier B.V., All rights reserved.