# University of Minnesota

Aerospace Engineering and Mechanics

**Spring 1997 Seminar Series**

*Adaptive and Backstepping Boundary Controllers for Flexible
Beams with Actuator Dynamics*

__Abstract__

**Hybrid systems of linear partial differential (PDE) and ordinary
differential (ODE) equations accurately model a variety of electromechanical
flexible systems. Models of smart structures, lightweight robots, magnetic
bearing rotors, and web/fiber handling machinery, for example, involve
distributed parameter material models and lumped parameter sensors and
actuators. High performance operation of these systems requires the development
of active feedback controllers that compensate for material flexibility. Due to
the lack of control design tools for hybrid ODE/PDE systems, however, many
researchers discretize the PDE subsystem using approximate methods such as
finite elements. Unfortunately, a controller that stabilizes the discretized
model may destabilize the high frequency modes of the actual distributed system
(i.e. spillover instability). Use of a large number of modes to approximate
the PDE subsystem often results in a complex, high-order control algorithm.
**

In this seminar, implementable, low order controllers are developed
for two hybrid ODE/PDE beam systems. First, boundary force controllers are
developed for a cantilevered Euler-Bernoulli beam with an end mass. An exact
model knowledge controller exponentially stabilizes the beam displacement given
the mechanical system parameters and measurements of the shear, shear-rate, and
velocity at the end of the beam. The exact model knowledge controller is then
redesigned as an adaptive controller that asymptotically stabilizes the beam
displacement while compensating for parametric uncertainty. Second, voltage
level controllers are derived for an electrically-driven, flexible link gantry
robot. The electrical subsystem dynamics for a permanent magnet brushed DC
motor couple with the distributed link dynamics. An integrator backstepping
boundary control law is developed that provides a stabilizing control force on
the flexible link using the motor voltage input. A velocity observer estimates
the gantry velocity, eliminating one feedback sensor. Experimental results
demonstrate the improved performance provided by the controllers.

###
Monday, April 21, 1997

209 Akerman Hall

2:30 p.m. - 3:30 p.m.

#### Refreshments served after each seminar in
227 AKERMAN HALL .

Disability accommodations provided upon request.

Contact
Leslie Petrus : Secretarial Assistant,
(612) 625-8000.