Backstepping Boundary Controllers and Observers for the Slender Timoshenko Beam: Part I - Design

In this paper we present the first extension of the backstepping methods that we have developed so far for control of parabolic PDEs (thermal, fluid, and chemical reaction dynamics) to second-order PDE systems (often referred loosely as hyperbolic) which model flexible structures and acoustics. We introduce controller and observer designs capable of adding damping to a model of beam dynamics using actuation only at the beam base and using sensing only at the beam tip. We present our designs for the Timoshenko beam model (the most advanced in the catalog of beam models, which also includes the simplest Euler-Bernoulli, as well as the Rayleigh and "shear" beam models) under the assumption that the beam is "slender." We allow the presence of a small amount of Kelvin-Voigt (KV) damping, which models internal material friction (rather than viscous interaction with the environment) and is present in every realistic material, though our method also applies in the completely undamped case. The closed-loop system with our backstepping boundary feedback is equivalent to a model of a string immersed in viscous fluid, with increased stiffness, supported on one end by a spring of high stiffness and on the other end by a damper. Such a closed loop system is very well damped and achieves the same excellent damping performance as the previous damping feedbacks which apply actuation at the free end of the beam. To ease the reader into the ideas, we first present the same method for a wave equation (string) with one free end and with a small amount of KV damping and then pursue the development for the Timoshenko beam model