Passive, Iterative, and Repetitive Control for Flexible Distributed Parameter Systems

Open Access
Zhao, Haiyu
Graduate Program:
Mechanical Engineering
Doctor of Philosophy
Document Type:
Date of Defense:
October 06, 2005
Committee Members:
  • Christopher Rahn, Committee Chair
  • Kon Well Wang, Committee Member
  • Qian Wang, Committee Member
  • Farhan Gandhi, Committee Member
  • Feedforward control
  • flexible distributed parameter systems
  • vibration control
  • BIBO stability
Many engineering structures have distributed parameter models governed by partial differential equations. Without damping, distributed flexible structures are not stable due to the infinite number of resonances at natural frequencies. Bounded sinusoidal inputs at these frequencies can cause unbounded response. This thesis shows that Passive Control, Iterative Learning Control (ILC), and Repetitive Learning Control (RLC) can be designed to reduce tracking or regulation errors in response to bounded, periodic inputs. Distributed flexible strings, beams, membranes, plates, axially moving materials, electrostatic microbridges, and flexible whisker contact imagers are studied. Passive control using distributed or boundary damping is proven to stabilize the response of strings, beams, membranes, and plates. Damping ensures bounded response to bounded distributed and boundary inputs. Distributed viscous or Kelvin-Voigt material damping can guarantee pointwise or strong boundedness for strings and beams and weak boundedness for membranes and plates. Translational damping on one boundary stabilizes strings and beams. Damping on part of the boundary can also weakly stabilize the forced response of membranes and plates, provided the damped and undamped boundary normals satisfy certain conditions. For example, damping on half and one side of the boundary is sufficient for circular and rectangular domains, respectively. Iterative Learning Control provides precise tension and speed control of axially moving material systems to enable high speed processing of paper, plastics, fibers, and films. PD tension/speed control is proven to ensure strong and weak boundedness of distributed displacement and tension, respectively, in a single span axially moving material system. ILC provides the same theoretical result with half the speed error and 30% of the tension error of PD control using the same control effort. Repetitive Learning Control is applied to an electrostatic microbridge and a repetitive contact imager. Electrostatic microactuators are used extensively in MEMS sensors, RF switches, and microfluidic pumps. Due to high bandwidth operation, however, reduction of residual vibration using feedback control is difficult to implement. Feedforward RLC is designed, proven stable, and simulated for an electrostatic microbridge under a periodic desired spatial/time trajectory. High residual stresses in the microbridge mean that bending stiffness can be neglected and a pinned string model with uniform loading is appropriate. Squeeze film damping ensures boundedness of the distributed transverse displacement. Offline RLC processing of the average displacement as measured by capacitive sensing updates a waveform generator's parameters. Simulations show a 36% reduction in midspan overshoot under repetitive control. Repetitive contact imaging uses a flexible whisker attached to a two axis robot through a load cell. Assuming small deformations and rotations, the pitch axis decouples from yaw. The yaw axis, under PD control, sweeps periodically back and forth across the object while the pitch axis, under RLC, maintains a uniform contact force. Once the RLC converges, the 3D contact points can be determined using an elastica algorithm. RLC is proven stable based on a distributed parameter beam model and experimentally shown to outperform PD control with 75% reduction in the moment error.