Motion planning and tracking for tip displacement and deflection angle for flexible beams

Explicit motion-planning reference solutions are presented for flexible beams with Kelvin-Voigt (KV) damping. The goal is to generate periodic reference signals for the displacement and deflection angle at the free-end of the beam using only actuation at the base. The explicit deflection angle reference solution is found as a result of writing the shear beam model in a strict-feedback form. Special "partial differential equation (PDE) backstepping" transformations relate the strict-feedback model to a "target system," governed by an exponentially stable wave equation with KV damping, whose displacement reference solution is relatively easy to find. The explicit beam displacement reference solution is found using the target system solution and an inverse backstepping transformation. The explicit reference solutions for the wave equation and shear beam with KV damping are novel results. State-feedback tracking boundary controllers are found by extending previous PDE backstepping stabilization results. Application of the shear beam results to the more complicated Timoshenko beam is discussed.