Vibration Control for a Three-Dimensional Variable Length Flexible String With Time-Varying Actuator Faults and Unknown Control Directions

Abstract

In this study, actuator faults and unknown control directions are taken into account concurrently for a three-dimensional variable length flexible string system modeled by partial differential equations (PDEs). Different from the unknown fault parameters being restricted to just positive constants in the previous fault-tolerance research on PDEs systems, the time-varying actuator faults are considered, which is more practical but challenging. Additionally, the control directions of the system are unknown which can also be different in three directions. To overcome the difficulties, adaptive control laws employing the Nussbaum function are designed, and some auxiliary control signals are intended for service. Under the proposed boundary control scheme, uniform ultimate boundedness of the closed-loop system is guaranteed, and the deflections of the flexible string can be suppressed to a small neighborhood of zero. Finally, the simulations are implemented to indicate the control effect. <i>Note to Practitioners</i>&#x2014;String structural systems are widely used in fields of indoor and outdoor industrial and mining enterprises, ports, wharves, and seabeds. To improve work efficiency and safety, effectively suppressing the vibration of the flexible string is a crucial problem to be solved. This paper proposes a novel adaptive boundary control scheme to reduce the deflections of the flexible string system subject to time-varying actuator faults and unknown control directions in three-dimensional space. The control scheme is designed without simplification, which can provide a theoretical basis and a feasible solution for the control of flexible string systems in practical engineering. In the future, we will address the modeling and control problems of moving three-dimensional variable length flexible string systems.