Modeling and vibration control for a flexible pendulum inverted system based on a PDE observer

ABSTRACT

This study addresses the problem of trajectory control of a flexible pendulum inverted system on the basis of the partial differential equation (PDE) and ordinary differential equation (ODE) dynamic model. One of the key contributions of this study is that a new model is proposed to simplify the complex system. In addition, this study proposed a nonlinear PDE observer to estimate distributed positions and velocities along flexible pendulum. Singular perturbation method is proposed to solve the coupling system of nonlinear PDE observer. The nonlinear PDE observer is divided into a fast subsystem and a slow subsystem by the use of the singular perturbation method. To stabilise this fast subsystem, a boundary controller is proposed at the free end of the beam. The sliding-mode control method is proposed to design controller for slow subsystems. The asymptotic stability of both the proposed nonlinear PDE observer and controller is validated by theoretical analysis. The results are illustrated by simulation.

KEYWORDS:

• Flexible pendulum inverted system
• PDE observer
• vibration control
• sliding-mode control
• boundary control

Additional information

Funding

This work was partially supported by the Research Fund for Doctoral Program of Higher Education of China [grant number 20121102110008] and the National Natural Science Foundation of China [grant number 61374048].