Infinite Dimensional Observers for Vibrating Systems

The paper considers linear systems on a Hilbert space with a skew-adjoint generator. The output space is assumed to be another Hilbert space and the system is exactly observable. We propose a Kalman type observer. We prove exponential stability of the proposed observer under the assumption of some system regularity and we estimate its decay rate. We demonstrate the applicability of our observer by working out the details of observer design for a rotating beam system. Using spectral analysis we determine exactly the exponential decay rate of the observer for this example. Based on the example we propose a method to assign arbitrarily the exponential decay rate for the constructed observer. A numerical simulation result will be presented to validate the observer for application