Compensating the Distributed Effect of a Wave PDE in the Actuation or Sensing Path of Multi-Input and Multi-Output LTI Systems

The problem of compensation of infinite-dimensional actuator or sensor dynamics of more complex type than pure delay was solved recently using the backstepping method for PDEs. In this paper we construct an explicit feedback law for a multi-input LTI system which compensates the wave PDE dynamics and stabilizes the overall system. Our design is based on a novel infinite-dimensional backstepping-forwarding transformation. We illustrate the effectiveness of our design with a simulation example of a single-input second order system, in which the wave input enters the system through two different channels, each one located at a different point in the domain of the wave PDE. Finally, we consider a dual problem where we design an exponentially convergent observer that compensates the distributed effect of the wave sensor dynamics.