Dynamic modeling and nonlinear boundary control of hybrid Euler–Bernoulli beam system with a tip mass

Abstract

This paper presents dynamic modeling and Lyapunov-based boundary control of a hybrid Euler–Bernoulli beam. The beam is hybrid in the sense that it holds both rigid and elastic motions. The beam is equipped with actuators hub at one end and it carries the payload as the tip mass at the free end. The distributed parameter dynamic equations (i.e. partial differential equations governing the hybrid beam motion) are derived using Hamilton’s principle. The dynamic model consists of four distributed parameter dynamic equations, representing the beam vibration and rigid motion in the plane, coupled to the discrete dynamic boundary equations. This paper uses the system equations to achieve model-based control laws that asymptotically stabilize the beam vibration while driving the rigid body position and orientation to the desired set point. The control system applies three forces/torque at the hub to regulate the rigid body position/orientation and a transverse force at the free end to suppress the beam vibration. In this regard, the control system only applies actuation and makes measurement at the beam boundary, thus excluding the need for distributed actuators or sensors. Furthermore, the proposed method directly uses the system equations for the control design without model truncation to rule out spillover instabilities. The closed-loop system stability is shown through Lyapunov-based analysis. Numerical simulations demonstrate the effectiveness of the proposed method.