Output feedback stabilization of an unstable wave equation with observations subject to time delay
In this paper we present the first extension of the backstepping methods developed for control of parabolic PDEs (modeling thermal, fluid, and chemical reaction dynamics, including Navier-Stokes equations and turbulence) to secondorder PDE systems (often referred loosely as hyperbolic) which model flexible structures and acoustic. 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. Interestingly, the backstepping method does not apply to the simplest Euler-Bernoulli model but does apply to more realistic models, including the Timoshenko beam model under the assumption that the beam is “slender.” For our method to be applicable it is necessary that the beam model includes a small amount of Kelvin-Voigt damping. Such damping models internal material friction (rather than viscous interaction with the environment) and is present in every realistic material. We don’t use the KV damping as a source of dissipation but as a means of controllability of the beam. With only a small amount of KV damping present in the uncontrolled system, we are able to introduce a substantial amount of damping of classical type (velocity-based). The closed-loop system with our boundary feedback included can be transformed into a form where both the added damping and an addition of “stiffness” are evident. As we show, this simultaneous change in damping and stiffness results in an overall shift of the eigenvalues to the left in the complex plane and in the improvement of the damping ratio of all the eigenvalues. To ease the reader into main concepts, we first present the same method for a wave equation with a small amount of KV damping and then pursue the development for a shear beam and Timoshenko beam model. The same result can be developed for the Rayleigh beam model which is structurally the same as the shear beam model but with different parameters. THE FINAL VERSION OF THE PAPER WILL INCLUDE SIMULATION RESULTS.