Boundary Control of a Two Dimensional Rotating Euler-Bernoulli Flexible Body-Beam System

Abstract:

We have developed boundary controllers for a rotating Euler-Bernoulli flexible body-beam system. Specifically, we develop an exact model knowledge controller which exponentially regulates the free-end mass displacement and the angular velocity setpoint tracking error, and an adaptive controller which exhibits asymptotic regulatiuon while compensating for parametric uncertainty. The experimental test stand consists of the following components:

  • A long flexible PVC shaft serves as the rotor, clamped at one end.
  • A Slip-Ring assembly at the fixed end allows electrical connections to be passed to the rotating beam.
  • A DC Motor applies the Torque to the clamped end of the beam through a pulley-transmission system.
  • A two-axis shear sensor located at the free end measures the shear force.
  • A magneto-ferrous disk at the free-end allows the magnetic bearing to apply a control force to the rotor.
  • A large air gap magnetic bearing serves as the actuator at the free end.
  • A high luminosity LED is placed at the center of the magneto-ferrous disk.
  • Two Linear CCD cameras are used to measure the beams free end displacement from the LED.

The controller was implemented on a Pentium 166 MHz computer running QNX, a real time operating system. The software used was Qmotor, a Graphical User interface developed in house which allows real-time control. This software has the unique ability to allow the user to change control gains without recompiling the program. The control program itself was written in C. The data aquisition was done through a MuliQ data acuisition board. Techron Linear Power Amplifiers are used to provide the current neccesary to drive the mortor and magnetic bearing assembly.

Experimental Setup

Theory

Over the last decade, the study of the effects of flexiblility on lightweight body-beam systems has been propelled into the limelight. Motivated, for example, by the prohibitive cost of placing equipment in space, many structural designers and control researchers have focused on lightweight mechanical systems with rigid and flexible components. Mathemtically, such systems are often characterized by a combination of ordinary differential equations (ODEs), partial differentialk equations (PDEs), and a set of boundary conditions. Often, there exists a dynamic coupling between the rigid and flexible components; which makes the design of high-performance controllers for such systems very complex.

In this experiment, we present a nonlinear boundary control law for the nonlinear, hybrid body-beam model. Specifically, we design an exact model knowledge controller which exponentially stabilizes the beam displacement while providing exponential regulation of the body-beam's angular speed. This control law requires the measurements of the following states at the free end:

  • The displacement of the beam,
  • The angular speed,
  • The shear force experienced by the beam,
  • And the rate of change of shear.
The control law is comprised of a control torque applied to the disk and a control force which is applied to the free end of the rotating body-beam. We then redesign the exact model knowledge controller to compensate for mechanical parametric uncertainties. The adaptive control law exhibits asymptotic stability.

Experimental results.