Quantized Time-Varying Fault-Tolerant Control for a Three-Dimensional Euler–Bernoulli Beam With Unknown Control Directions

In this study, quantization and time-varying actuator faults are considered for an uncertain three-dimensional Euler-Bernoulli beam system. The uncertainty is shown in that the control directions of the three actuators for the beam system may be unknown in practice. The three-dimensional Euler-Bernoulli beam is modeled as a distributed parameter system, where the system’s governing equations are represented by a set of partial differential equations (PDEs) and boundary equations by ordinary differential equations (ODEs). Quantization is a necessary process in the network control field, and the quantized control algorithm is proposed and realized by logarithmic quantizers in this paper. Based on quantized control, the adaptive time-varying actuator fault-tolerant control scheme is designed with the aid of the Nussbaum functions and some auxiliary signals, which can also compensate for unknown control directions skillfully. Under the designed control scheme, all the signals are guaranteed to be bounded and the vibration of the beam is controlled. Finally, the simulation is done to verify the control effect.
Note to Practitioners
—The Euler-Bernoulli beam is used to model many practical mechanical systems, such as flexible manipulators, flexible satellite panels, and flexible marine risers. Therefore, studies on the Euler-Bernoulli beam are fundamental and essential. Furthermore, although the three-dimensional models based on PDEs of the Euler-Bernoulli beam have stronger nonlinearity and coupling, it is more aligned with realistic working conditions. The engineering problems studied in this paper include unknown control directions, input quantization, and time-varying actuator faults. Using the boundary control strategy, adaptive control method, and some auxiliary systems along with the Nussbaum functions, the system stability is proven without any simplification, and the control goals are achieved. In the future, the other kind of beam, the Timoshenko beam which has more complex dynamics can be devoted to doing some research.