UC San Diego

The problems of vibration suppression, stabilization, motion planning, and tracking for flexible beams are important for many practical mechanical systems, such as the cantilevered beam in an atomic force microscope, a solar panel array on a satellite, the boom on a crane, or a structure mounted on a shake table. This work approaches the aforementioned control design problems using two methods; backstepping for partial differential equations (PDEs), and extremum seeking. PDE backstepping is a form of model reference control for infinite dimensional systems where boundary control is used to make a closed- loop system perform like a target system, which is the reference model with desirable performance characteristics. Infinite dimensional state transformations, relating plant and target systems, are the key to PDE backstepping designs and are used to find the desired boundary controllers. PDE backstepping techniques are presented for motion planning and tracking for the string and shear beam with Kelvin-Voigt (KV) damping, and a combination of PDE backstepping and gain scheduling is presented for extending linear PDE backstepping techniques to stabilization, motion planning, and tracking for the string and shear beam with KV damping and boundary-displacement dependent free-end nonlinearities. Explicit motion planning reference solutions for the shear beam are found, using backstepping transformations, as a function of the target system reference solution, which itself is a function of the string solution. The string solution is the easiest to find, and is done using traditional motion planning tools. Tracking controllers are found as a combination of vibration suppression and motion planning controllers. Gain scheduling is a method that replaces nonlinear control design with the design of a family of linear controllers. When combined with linear PDE backstepping techniques, it provides a means of extending vibration suppression, stabilization, motion planning and tracking results to the string and shear beam with free-end nonlinearities. Gain scheduling based designs, which produce locally stabilizing controllers parameterized by a function of the nonlinearity, are a more simple and manageable alternative to full-state feedback linearizing nonlinear PDE control designs. Extremum seeking (ES) is a non-model based method used for tuning parameter(s) to optimize an unknown nonlinear map. ES tuning for positive- position-feedback compensators is presented as a method of selecting the parameter(s) of the compensator to improve vibration suppression on a mechanical system forced by a persistent sinusoidal disturbance. ES tuning for motion planning and tracking is presented as a method for tuning the amplitude and phase of a sinusoidal input to a stable linear plant to generate and track a sinusoid of desired amplitude and phase at the output