Adaptive Estimation of a Flexible Beam

An on-line or adaptive identification scheme for a flexible beam and, more generally, second order in time infinite dimensional or distributed parameter systems, is constructed. The estimator takes the form of an infinite dimensional initial value problem for a linear evolution system whose state consists of an estimate for the plant and an estimate for the parameters. A Lyapunov estimate is used to argue state convergence. Parameter convergence is argued via the extension of the finite dimensional notion of richness or persistence of excitation. A laboratory experiment involving an aluminum beam with piezoceramic actuation at the root and proximity displacement sensors mounted alongside the beam is discussed. The purpose of the experiment is to test the feasibility of the identification scheme.