Periodic Solutions and Their Regions of Attraction for Flexible Structures Under Relay Feedback Control With Nonlinear Control Law

A method for calculating all periodic solutions and their domains of attraction for flexible systems under nonlinear feedback control is presented. The systems considered consist of mechanical systems with many flexible modes and a relay type controller coupled with a nonlinear control law operating in a feedback configuration. The proposed approach includes three steps. First, limit cycle frequencies and periodic fixed points are computed exactly, using a block diagonal state-space modal representation of the plant dynamics. Then the relay switching surface is chosen as the Poincare mapping surface and is discretized using the cell mapping method. Finally, the region of attraction for each limit cycle is computed using the cell mapping algorithm and employing an error based convergence criterion. An example consisting of a system with two modes, a relay with dead-zone and hysteresis, and a nonlinear control law with a signed velocity squared term is used to demonstrate the proposed approach.